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Problems Set 5: Graph Theory

1 Graph Theory

1. Show that every connected graph has at least two verticesu and v such thatGuand Gv

are connected.

2. Show that for any graphG = (V, E) the vertex setVcan be partitioned into two sets V1 and

V2 such that

e(V1) +e(V2)

|E|

2

wheree(Vi) means the number of edges in Ewith both end points in Vi.

3. Prove that a regular bipartite graph of degree at least 2 does not contain a bridge.

4. Let G be a graph with minimum degree 2. Show that there exist a connected graph with

same degree sequence.

5. Let T1, . . . , T k be subtrees of a tree Tsuch that for all i, j the trees Ti and Tj have a vertex

in common. Show that Thas a vertex that is in all Ti.

6. LetG be a planar graph, with edges colored redand blue. Show that there is a vertex v such

that going round the vertex in a clockwise direction we encountered no more than two change

of colors.

7. IfG = (V, E) is a graph on nvertices such that all the vertices have even degree. Show that

the edge set Ecan be partitioned into pairwise disjoint sets C1, C2, . . . , C k such that for all

1 i k the subgraphs (V, Ci) is a cycle and a collection of isolated vertices.

8. If a graph has maximum degree less than or equal to k then it is (k+ 1)-colorable.

9. A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in

an undirected complete graph. That is, it is a directed graph in which every pair of vertices

is connected by a single directed edge. Prove that a tournament always has a Hamiltonianpath.

10. If in a directed graph number of incoming edges is equal to number of outgoing edges then

the graph has an Eulerian Path.

11. Prove that then-dimensional cube graph has a Hamiltonian path.

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12. Prove that 101 999 grid does not have a Hamiltonian Cycle.

13. Prove that a graph with n vertices and n+ 2 edges in planar.

14. How many 4-cycles are in Kn,n?

15. Prove that eitherG or G is connected.

16. A chord of a cycle is an edge that connects two non-adjacent vertices in the cycle. Prove that

if every node ofG has degree 3 then Gcontains a cycle with a chord.

17. IfG is a bipartite graph with m nodes on each side. If each node has degree more than m/2

then prove that it has a perfect matching.

18. Prove that a planar bipartite graph has at most 2n 4 edges.

19. If every node has degree dand at least one vertex has degree strictly less than d then prove

that the graph is d-colorable.

20. If every face of a planar graph has even number of edges then prove that the graph is bipartite.

21. Prove that there is a tournamentT withn players and at leastn!2(n1) Hamiltonian paths.

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