locally injective homomorphisms · gary macgillivray university of victoria victoria, bc, canada...
TRANSCRIPT
![Page 1: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/1.jpg)
Locally injective homomorphisms
Gary MacGillivray
University of VictoriaVictoria, BC, Canada
![Page 2: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/2.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 3: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/3.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 4: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/4.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 5: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/5.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 6: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/6.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 7: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/7.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 8: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/8.jpg)
Homomorphisms
For graphs G and H, think of V (H) as a set of colours. ColourV (G ) so that adjacent vertices get adjacent colours.
G H
I A homomorphism G → H is a function f : V (G )→ V (H)such that f (x)f (y) ∈ E (H) whenever xy ∈ E (G ).
I If H ∼= Kn, then a homomorphism G → H is an n-colouring ofG .
![Page 9: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/9.jpg)
Locally injective homomorphisms
A homomorphism G → H is locally injective if its restriction toN(x) is injective, for every x ∈ V (G ).
G H
When H ∼= Kn, a locally injective homomorphism G → H is alocally injective proper n-colouring.
![Page 10: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/10.jpg)
Locally injective homomorphisms
A homomorphism G → H is locally injective if its restriction toN(x) is injective, for every x ∈ V (G ).
G H
When H ∼= Kn, a locally injective homomorphism G → H is alocally injective proper n-colouring.
![Page 11: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/11.jpg)
Locally injective homomorphisms
A homomorphism G → H is locally injective if its restriction toN(x) is injective, for every x ∈ V (G ).
G H
When H ∼= Kn, a locally injective homomorphism G → H is alocally injective proper n-colouring.
![Page 12: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/12.jpg)
Locally injective homomorphisms
A homomorphism G → H is locally injective if its restriction toN(x) is injective, for every x ∈ V (G ).
G H
When H ∼= Kn, a locally injective homomorphism G → H is alocally injective proper n-colouring.
![Page 13: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/13.jpg)
Locally injective homomorphisms
A homomorphism G → H is locally injective if its restriction toN(x) is injective, for every x ∈ V (G ).
G H
When H ∼= Kn, a locally injective homomorphism G → H is alocally injective proper n-colouring.
![Page 14: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/14.jpg)
Locally injective homomorphisms
A homomorphism G → H is locally injective if its restriction toN(x) is injective, for every x ∈ V (G ).
G H
When H ∼= Kn, a locally injective homomorphism G → H is alocally injective proper n-colouring.
![Page 15: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/15.jpg)
Locally injective proper n-colourings: I
Properly colours the vertices so that no two vertices with acommon neighbour get the same colour (the colouring is injectiveon neighbourhoods).
I Colourings of the square.
(Join vertices at distance 2.)
I ∆ + 1 colours needed; ∆2 + 1 colours suffice.
![Page 16: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/16.jpg)
Locally injective proper n-colourings: I
Properly colours the vertices so that no two vertices with acommon neighbour get the same colour (the colouring is injectiveon neighbourhoods).
I Colourings of the square. (Join vertices at distance 2.)
I ∆ + 1 colours needed; ∆2 + 1 colours suffice.
![Page 17: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/17.jpg)
Locally injective proper n-colourings: I
Properly colours the vertices so that no two vertices with acommon neighbour get the same colour (the colouring is injectiveon neighbourhoods).
I Colourings of the square. (Join vertices at distance 2.)
I ∆ + 1 colours needed; ∆2 + 1 colours suffice.
![Page 18: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/18.jpg)
Locally injective proper n-colourings: I
Properly colours the vertices so that no two vertices with acommon neighbour get the same colour (the colouring is injectiveon neighbourhoods).
I Colourings of the square.
(Join vertices at distance 2.)
I ∆ + 1 colours needed; ∆2 + 1 colours suffice.
![Page 19: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/19.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].I There is a dichotomy in the list version [Fiala & Kratochvıl,
2006].
![Page 20: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/20.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].I There is a dichotomy in the list version [Fiala & Kratochvıl,
2006].
![Page 21: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/21.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].I There is a dichotomy in the list version [Fiala & Kratochvıl,
2006].
![Page 22: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/22.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].I There is a dichotomy in the list version [Fiala & Kratochvıl,
2006].
![Page 23: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/23.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].I There is a dichotomy in the list version [Fiala & Kratochvıl,
2006].
![Page 24: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/24.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].
I There is a dichotomy in the list version [Fiala & Kratochvıl,2006].
![Page 25: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/25.jpg)
Locally injective proper n-colourings: II
I Polynomial to decide if n ≤ 3 colours suffice; NP-complete forn ≥ 4 [Fiala & Kratochvıl, 2002].
I 2 colours suffice if and only if P3 is not a subgraph of G .
I 3 colours suffice if and only if neither K1,3 nor any cycle oflength not a multiple of 3 is a subgraph of G .
I Not much is known about the complexity of injectivehomomorphisms to irreflexive graphs.
I Polynomial when restricted to graphs of bounded treewidth (byCourcelle’s Theorem).
I There is a dichotomy for theta graphs [Lidicky & Tesar, 2011].I There is a dichotomy in the list version [Fiala & Kratochvıl,
2006].
![Page 26: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/26.jpg)
Locally injective homomorphisms to reflexive graphs: I
I A graph is reflexive if it has a loop at every vertex.
I If H is reflexive and G → H is a homomorphism, adjacentvertices of G can have the same “colour” (image), even in aninjective homomorphism.
reflexive HG
I When H ∼= Kn, a locally injective homomorphism G → H is alocally injective improper n-colouring
![Page 27: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/27.jpg)
Locally injective homomorphisms to reflexive graphs: I
I A graph is reflexive if it has a loop at every vertex.
I If H is reflexive and G → H is a homomorphism, adjacentvertices of G can have the same “colour” (image), even in aninjective homomorphism.
reflexive HG
I When H ∼= Kn, a locally injective homomorphism G → H is alocally injective improper n-colouring
![Page 28: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/28.jpg)
Locally injective homomorphisms to reflexive graphs: I
I A graph is reflexive if it has a loop at every vertex.
I If H is reflexive and G → H is a homomorphism, adjacentvertices of G can have the same “colour” (image), even in aninjective homomorphism.
reflexive HG
I When H ∼= Kn, a locally injective homomorphism G → H is alocally injective improper n-colouring
![Page 29: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/29.jpg)
Locally injective improper n-colourings: I
I Polynomial to decide if k ≤ 2 colours suffice; NP-complete ifk ≥ 3 [Hahn, Kratochvıl, Siran, & Sotteau, 2002].
I 2 colours suffice if and only if neither K1,3 nor an odd cyclesubgraph of G .
I ∆ colours needed; ∆2 −∆ + 1 colours suffice.
I Polynomial for any fixed n when restricted chordal graphs[Hell, Raspaud, & Stacho, 2008].
I Lots of results known for for planar graphs, for example:I ∆ + 2 colours suffice when girth g ≥ 7
[Dimitrov, Luzar, & Skrekovski, 2009].I ∆ + 1 colours suffice when girth g ≥ 7 and ∆ ≥ 7
[Bu, & Lu, 2012].I ∆ colours suffice when girth g ≥ 7 and ∆ ≥ 16
[Borodin, & Ivanova, 2011].
I Not much is known about the complexity of injectivehomomorphisms to reflexive graphs.
![Page 30: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/30.jpg)
Locally injective improper n-colourings: I
I Polynomial to decide if k ≤ 2 colours suffice; NP-complete ifk ≥ 3 [Hahn, Kratochvıl, Siran, & Sotteau, 2002].
I 2 colours suffice if and only if neither K1,3 nor an odd cyclesubgraph of G .
I ∆ colours needed; ∆2 −∆ + 1 colours suffice.
I Polynomial for any fixed n when restricted chordal graphs[Hell, Raspaud, & Stacho, 2008].
I Lots of results known for for planar graphs, for example:I ∆ + 2 colours suffice when girth g ≥ 7
[Dimitrov, Luzar, & Skrekovski, 2009].I ∆ + 1 colours suffice when girth g ≥ 7 and ∆ ≥ 7
[Bu, & Lu, 2012].I ∆ colours suffice when girth g ≥ 7 and ∆ ≥ 16
[Borodin, & Ivanova, 2011].
I Not much is known about the complexity of injectivehomomorphisms to reflexive graphs.
![Page 31: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/31.jpg)
Locally injective improper n-colourings: I
I Polynomial to decide if k ≤ 2 colours suffice; NP-complete ifk ≥ 3 [Hahn, Kratochvıl, Siran, & Sotteau, 2002].
I 2 colours suffice if and only if neither K1,3 nor an odd cyclesubgraph of G .
I ∆ colours needed; ∆2 −∆ + 1 colours suffice.
I Polynomial for any fixed n when restricted chordal graphs[Hell, Raspaud, & Stacho, 2008].
I Lots of results known for for planar graphs, for example:I ∆ + 2 colours suffice when girth g ≥ 7
[Dimitrov, Luzar, & Skrekovski, 2009].I ∆ + 1 colours suffice when girth g ≥ 7 and ∆ ≥ 7
[Bu, & Lu, 2012].I ∆ colours suffice when girth g ≥ 7 and ∆ ≥ 16
[Borodin, & Ivanova, 2011].
I Not much is known about the complexity of injectivehomomorphisms to reflexive graphs.
![Page 32: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/32.jpg)
Locally injective improper n-colourings: I
I Polynomial to decide if k ≤ 2 colours suffice; NP-complete ifk ≥ 3 [Hahn, Kratochvıl, Siran, & Sotteau, 2002].
I 2 colours suffice if and only if neither K1,3 nor an odd cyclesubgraph of G .
I ∆ colours needed; ∆2 −∆ + 1 colours suffice.
I Polynomial for any fixed n when restricted chordal graphs[Hell, Raspaud, & Stacho, 2008].
I Lots of results known for for planar graphs, for example:I ∆ + 2 colours suffice when girth g ≥ 7
[Dimitrov, Luzar, & Skrekovski, 2009].I ∆ + 1 colours suffice when girth g ≥ 7 and ∆ ≥ 7
[Bu, & Lu, 2012].I ∆ colours suffice when girth g ≥ 7 and ∆ ≥ 16
[Borodin, & Ivanova, 2011].
I Not much is known about the complexity of injectivehomomorphisms to reflexive graphs.
![Page 33: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/33.jpg)
Locally injective improper n-colourings: I
I Polynomial to decide if k ≤ 2 colours suffice; NP-complete ifk ≥ 3 [Hahn, Kratochvıl, Siran, & Sotteau, 2002].
I 2 colours suffice if and only if neither K1,3 nor an odd cyclesubgraph of G .
I ∆ colours needed; ∆2 −∆ + 1 colours suffice.
I Polynomial for any fixed n when restricted chordal graphs[Hell, Raspaud, & Stacho, 2008].
I Lots of results known for for planar graphs, for example:I ∆ + 2 colours suffice when girth g ≥ 7
[Dimitrov, Luzar, & Skrekovski, 2009].I ∆ + 1 colours suffice when girth g ≥ 7 and ∆ ≥ 7
[Bu, & Lu, 2012].I ∆ colours suffice when girth g ≥ 7 and ∆ ≥ 16
[Borodin, & Ivanova, 2011].
I Not much is known about the complexity of injectivehomomorphisms to reflexive graphs.
![Page 34: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/34.jpg)
Locally injective improper n-colourings: I
I Polynomial to decide if k ≤ 2 colours suffice; NP-complete ifk ≥ 3 [Hahn, Kratochvıl, Siran, & Sotteau, 2002].
I 2 colours suffice if and only if neither K1,3 nor an odd cyclesubgraph of G .
I ∆ colours needed; ∆2 −∆ + 1 colours suffice.
I Polynomial for any fixed n when restricted chordal graphs[Hell, Raspaud, & Stacho, 2008].
I Lots of results known for for planar graphs, for example:I ∆ + 2 colours suffice when girth g ≥ 7
[Dimitrov, Luzar, & Skrekovski, 2009].I ∆ + 1 colours suffice when girth g ≥ 7 and ∆ ≥ 7
[Bu, & Lu, 2012].I ∆ colours suffice when girth g ≥ 7 and ∆ ≥ 16
[Borodin, & Ivanova, 2011].
I Not much is known about the complexity of injectivehomomorphisms to reflexive graphs.
![Page 35: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/35.jpg)
Locally injective colourings
LetI χs(G ) = min number of colours in a locally injective proper
colouring, andI χi (G ) = min number of colours in a locally injective improper
colouring.
Theoremχi ≤ χs ≤ 2χi .
[Kim & Oum, 2009]
![Page 36: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/36.jpg)
Locally injective colourings
LetI χs(G ) = min number of colours in a locally injective proper
colouring, andI χi (G ) = min number of colours in a locally injective improper
colouring.
Theoremχi ≤ χs ≤ 2χi .
[Kim & Oum, 2009]
![Page 37: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/37.jpg)
Locally injective colourings
LetI χs(G ) = min number of colours in a locally injective proper
colouring, andI χi (G ) = min number of colours in a locally injective improper
colouring.
Theoremχi ≤ χs ≤ 2χi .
[Kim & Oum, 2009]
![Page 38: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/38.jpg)
Locally injective colourings
LetI χs(G ) = min number of colours in a locally injective proper
colouring, andI χi (G ) = min number of colours in a locally injective improper
colouring.
Theoremχi ≤ χs ≤ 2χi .
[Kim & Oum, 2009]
![Page 39: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/39.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 40: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/40.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 41: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/41.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 42: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/42.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 43: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/43.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 44: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/44.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 45: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/45.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 46: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/46.jpg)
Homomorphisms between oriented graphs
Colour the vertices so that if u is adjacent to v , then the colour ofu is adjacent to the colour of v .
G H
What should injective mean?
![Page 47: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/47.jpg)
Injective homomorphisms: the options
There are 3 possible definitions of injective for homomorphisms oforiented graphs.
1. injective on in-neighbourhoods only
2. injective on in- and out-neighbourhoods separately
3. injective on in- and out-neighbourhoods together
![Page 48: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/48.jpg)
Injective homomorphisms: the options
There are 3 possible definitions of injective for homomorphisms oforiented graphs.
1. injective on in-neighbourhoods only
2. injective on in- and out-neighbourhoods separately
3. injective on in- and out-neighbourhoods together
![Page 49: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/49.jpg)
Injective homomorphisms: the options
There are 3 possible definitions of injective for homomorphisms oforiented graphs.
1. injective on in-neighbourhoods only
2. injective on in- and out-neighbourhoods separately
3. injective on in- and out-neighbourhoods together
![Page 50: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/50.jpg)
Injective homomorphisms: the options
There are 3 possible definitions of injective for homomorphisms oforiented graphs.
1. injective on in-neighbourhoods only
2. injective on in- and out-neighbourhoods separately
3. injective on in- and out-neighbourhoods together
1. is the most extensively studied [Swarts, 2008].
I When the target oriented graph H is reflexive, there is adichotomy.
I When the target oriented graph is irreflexive (no loops), thecomplexity is at least as rich as for all digraph homomorphismproblems.
![Page 51: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/51.jpg)
Injective homomorphisms: the options
There are 3 possible definitions of injective for homomorphisms oforiented graphs.
1. injective on in-neighbourhoods only
2. injective on in- and out-neighbourhoods separately
3. injective on in- and out-neighbourhoods together
2. and 3. We know the complexity for all tournaments on smallnumbers of vertices, and in several infinite familiies. In both cases
I When the target tournament H is reflexive, the problem ispolynomial if |V (H)| ≤ 2, and NP-complete if |V (H)| = 3.
I When the target tournament H is irreflexive, the problem ispolynomial if |V (H)| ≤ 3, and NP-complete if |V (H)| = 4.
[Campbell, Clarke, & GM, 2011]
![Page 52: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/52.jpg)
Injective oriented n-colouringsOriented n-colouring ≡ homomorphism to some oriented graphon n vertices.
I The target oriented graphs can be reflexive or irreflexive, sothere are 6 possible injective oriented colourings.
I In the case of proper colourings, injectivity on in- andout-neighbourhoods separately is same as on them together(vertices joined by a directed path of length 2 must getdifferent colours), so really there are 5 possibilities.
I The improper colourings are all Polynomial when n ≤ 2, andNP-complete when n ≥ 3. [Campbell, 2009; Clarke and GM,2011; GM, Raspaud and Swarts, 2013]
I The proper colourings are all polynomial when n ≤ 3, andNP-complete when n ≥ 4. [Clarke and GM, 2011; GM,Raspaud and Swarts, 2009, 2011]
I A description of the oriented graphs that are colourable canbe obtained in the Polynomial cases (a touch ugly).
![Page 53: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/53.jpg)
Injective oriented n-colouringsOriented n-colouring ≡ homomorphism to some oriented graphon n vertices.
I The target oriented graphs can be reflexive or irreflexive, sothere are 6 possible injective oriented colourings.
I In the case of proper colourings, injectivity on in- andout-neighbourhoods separately is same as on them together(vertices joined by a directed path of length 2 must getdifferent colours), so really there are 5 possibilities.
I The improper colourings are all Polynomial when n ≤ 2, andNP-complete when n ≥ 3. [Campbell, 2009; Clarke and GM,2011; GM, Raspaud and Swarts, 2013]
I The proper colourings are all polynomial when n ≤ 3, andNP-complete when n ≥ 4. [Clarke and GM, 2011; GM,Raspaud and Swarts, 2009, 2011]
I A description of the oriented graphs that are colourable canbe obtained in the Polynomial cases (a touch ugly).
![Page 54: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/54.jpg)
Injective oriented n-colouringsOriented n-colouring ≡ homomorphism to some oriented graphon n vertices.
I The target oriented graphs can be reflexive or irreflexive, sothere are 6 possible injective oriented colourings.
I In the case of proper colourings, injectivity on in- andout-neighbourhoods separately is same as on them together(vertices joined by a directed path of length 2 must getdifferent colours), so really there are 5 possibilities.
I The improper colourings are all Polynomial when n ≤ 2, andNP-complete when n ≥ 3. [Campbell, 2009; Clarke and GM,2011; GM, Raspaud and Swarts, 2013]
I The proper colourings are all polynomial when n ≤ 3, andNP-complete when n ≥ 4. [Clarke and GM, 2011; GM,Raspaud and Swarts, 2009, 2011]
I A description of the oriented graphs that are colourable canbe obtained in the Polynomial cases (a touch ugly).
![Page 55: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/55.jpg)
Injective oriented n-colouringsOriented n-colouring ≡ homomorphism to some oriented graphon n vertices.
I The target oriented graphs can be reflexive or irreflexive, sothere are 6 possible injective oriented colourings.
I In the case of proper colourings, injectivity on in- andout-neighbourhoods separately is same as on them together(vertices joined by a directed path of length 2 must getdifferent colours), so really there are 5 possibilities.
I The improper colourings are all Polynomial when n ≤ 2, andNP-complete when n ≥ 3. [Campbell, 2009; Clarke and GM,2011; GM, Raspaud and Swarts, 2013]
I The proper colourings are all polynomial when n ≤ 3, andNP-complete when n ≥ 4. [Clarke and GM, 2011; GM,Raspaud and Swarts, 2009, 2011]
I A description of the oriented graphs that are colourable canbe obtained in the Polynomial cases (a touch ugly).
![Page 56: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/56.jpg)
Injective oriented n-colouringsOriented n-colouring ≡ homomorphism to some oriented graphon n vertices.
I The target oriented graphs can be reflexive or irreflexive, sothere are 6 possible injective oriented colourings.
I In the case of proper colourings, injectivity on in- andout-neighbourhoods separately is same as on them together(vertices joined by a directed path of length 2 must getdifferent colours), so really there are 5 possibilities.
I The improper colourings are all Polynomial when n ≤ 2, andNP-complete when n ≥ 3. [Campbell, 2009; Clarke and GM,2011; GM, Raspaud and Swarts, 2013]
I The proper colourings are all polynomial when n ≤ 3, andNP-complete when n ≥ 4. [Clarke and GM, 2011; GM,Raspaud and Swarts, 2009, 2011]
I A description of the oriented graphs that are colourable canbe obtained in the Polynomial cases (a touch ugly).
![Page 57: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/57.jpg)
Summary ... a.k.a. the last slide
I Lots is known about injective colouring problems, and there islots left to do.
I What’s the story with (orientations of) planar graphs?
I Very little is known about the corresponding homomorphismproblems.
I For oriented graphs, and injectivity on in- andout-neighbourhoods separately or together, what is thecomplexity of injective homomorphism to a given (reflexive)tournament? Is there a dichotomy?
I Thank you for listening, reading, and not
throwing tomatoes.
![Page 58: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/58.jpg)
Summary ... a.k.a. the last slide
I Lots is known about injective colouring problems, and there islots left to do.
I What’s the story with (orientations of) planar graphs?
I Very little is known about the corresponding homomorphismproblems.
I For oriented graphs, and injectivity on in- andout-neighbourhoods separately or together, what is thecomplexity of injective homomorphism to a given (reflexive)tournament? Is there a dichotomy?
I Thank you for listening, reading, and not
throwing tomatoes.
![Page 59: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/59.jpg)
Summary ... a.k.a. the last slide
I Lots is known about injective colouring problems, and there islots left to do.
I What’s the story with (orientations of) planar graphs?
I Very little is known about the corresponding homomorphismproblems.
I For oriented graphs, and injectivity on in- andout-neighbourhoods separately or together, what is thecomplexity of injective homomorphism to a given (reflexive)tournament? Is there a dichotomy?
I Thank you for listening, reading, and not
throwing tomatoes.
![Page 60: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/60.jpg)
Summary ... a.k.a. the last slide
I Lots is known about injective colouring problems, and there islots left to do.
I What’s the story with (orientations of) planar graphs?
I Very little is known about the corresponding homomorphismproblems.
I For oriented graphs, and injectivity on in- andout-neighbourhoods separately or together, what is thecomplexity of injective homomorphism to a given (reflexive)tournament? Is there a dichotomy?
I Thank you for listening, reading, and not
throwing tomatoes.
![Page 61: Locally injective homomorphisms · Gary MacGillivray University of Victoria Victoria, BC, Canada gmacgill@uvic.ca. Homomorphisms For graphs G and H, think of V(H) as a set of colours](https://reader035.vdocuments.us/reader035/viewer/2022071022/5fd71260dae64e46127d4e4a/html5/thumbnails/61.jpg)
Summary ... a.k.a. the last slide
I Lots is known about injective colouring problems, and there islots left to do.
I What’s the story with (orientations of) planar graphs?
I Very little is known about the corresponding homomorphismproblems.
I For oriented graphs, and injectivity on in- andout-neighbourhoods separately or together, what is thecomplexity of injective homomorphism to a given (reflexive)tournament? Is there a dichotomy?
I Thank you for listening, reading, and not
throwing tomatoes.