4-chromatic projective graphs

4-C h ro mat ic Projective Graphs

~

D.A. Youngs VODAFONE LTD., NEWBURY

BERKS, UNlTED KlNGDOM e-mad [email protected]

ABSTRACT

We construct a family of 4-chromatic graphs which embed on the projective plane, and characterize the edge-critical members. The family includes many well known graphs, and also a new sequence of graphs, which serve to improve Gallai’s bound on the length of the shortest odd circuit in a 4-chromatic graph. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

We present a technique for constructing a large family of 4-chromatic graphs which embed on the projective plane, and also derive easy to verify conditions enabling us to determine the edge-critical members of this family. We refer to these graphs as “4-skeletons.”

This family includes various existing graphs such as the 4-chromatic Mycielski/Grotzsch graphs [5,8], the odd wheels, the Chvital graph [ I ] , and a lesser known family due to Gallai [lo] which are related to the 2-dimensional case of Sperner’s Simplex Lemma [131.

The method we use to show the 4-chromaticity of 4-skeletons is a particular instance of that used in [ 121 to produce graphs which cannot be oriented as Hasse Diagrams of Partially Ordered Sets. It follows implicitly that the graphs constructed here satisfy the stronger condition of being non-covering (or non-Hassr).

The next section describes the construction of 4-skeletons and states some preparatory results, whilst Section 3 discusses their chromatic properties and gives some examples.

In Section 4 we produce a sequence of graphs which do not contain short circuits of odd length. These are of interest to a problem of ErdZis [3] concerning the maximum of the length of the shortest odd circuit in a 4-chromatic graph.

In the final section we discuss some possible extensions of the methods developed here towards constructing highly chromatic sparse graphs.

The remainder of the section is devoted to introducing the necessary notation. For the sake of simplicity, we shall assume throughout that all graphs are connected. We follow Tutte [151 and utilize the theory of chain-groups for describing terms such as

circuit and walk. We merely present the essentials of the theory here, and refer the reader to [I51 for a full account.

Journal of Graph Theory, Vol. 21, No. 2, 219-227 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0364-9024/96/020219-09

220 JOURNAL OF GRAPH THEORY

Given a set S, a chain (to Z ) is a mapping f : S - Z. For s E S , f ( s ) is called the coefficient of s in ,f. Addition and integer multiplication can be applied to chains by simply adding or multiplying the individual coefficients respectively. The support of a chain is the subset of elements of S whose coefficients are non-zero. The chain having empty support is called the trivial chain. A chain-group is a nonempty set of chains that is closed with respect to addition and integer multiplication.

A 0-chain in a graph G is a mapping f : V ( G ) - Z, and a 1-chain is a mapping f : E(G) - Z . Let R be an orientation of G (this is merely a reference orientation). For each vertex u and directed edge d of G we define the incidence number i (u , d ) as follows: if d is a loop or is not incident with u then i ( u , d ) = 0, otherwise we set i ( u , d ) = 1 or i ( v , d ) = - 1, according to whether u is the head or the tail of d. For any 1-chain f of G we now define the bouiidary o f f as the 0-chain S f whose coefficients are defined as:

A I-chain whose boundary has (at most) two non-zero coefficients, equal to 21 is called a walk. Similarly, a 1-chain whose boundary is the trivial 0-chain we call a circuit. The set of all circuits form a chain-group. A circuit is called simple if its support does not properly contain the support of any other non-trivial circuit.

Notice that our definitions of walk and circuit agree with common usage of the terms, and in particular, that we may associate a “direction of traverse” with any walk or circuit. This “direction” along an edge is from tail to head (in the reference orientation) if the coefficient of the edge is positive, and from head to tail otherwise. We may describe walks and circuits in terms of the edges contained in their support (or rather the endpoints of these edges) and their corresponding coefficients. This is achieved by ordering the edges (treating those with coefficient greater that 1 in absolute value separately) according to the direction of traverse around the circuit (or walk). For convenience, this is how we shall describe walks and circuits in the sequel.

We end this section by remarking that the initial orientation R chosen above is a superficial device introduced purely for convenience. It has no effect on the subsequent definitions of terms such as walk or circuit.

2. 4-SKELETONS

Suppose G is a graph embedded on some surface M . With each face of G we may associate two face circuits one traversed in either direction. A set of face circuits is called complete if it contains precisely one face circuit corresponding to each face.

If G is a plane graph , then a face circuit enclosing the unbounded face of G we call a border circuit of G and denote it by B(G).

It is well known that a projective graph (i.e., a graph embedded on the projective plane) possesses simple circuits of two distinct types-those which separate the projective plane, and those which do not. We refer to these as base circuits and twisted circuits, respectively. For convenience, we define these circuits in the following slightly obscure terms:

Suppose that {Cl, CZ, . . . , Ck} is a complete set of face circuits of a projective graph G. Then we call a simple circuit C of G a base circuit if it may be written as a sum

4-CHROMATIC PROJECTIVE GRAPHS 221

k

c = -&jCj i = 1

where each ui satisfies

( Y j E {- l ,O,+l} .

Alternatively, we call C a twisted circuir if it may be written as a sum

k

2 c = &jCj i = l

where each ( ~ i satisfies

a; E { - l , O , +1}.

We call a graph G a 4-skeleton if i t satisfies the following conditions:

( I ) G is non-bipartite; ( 2 ) G has an embedding on the projective plane in which each face circuit has length 4.

Suppose we have a 4-skeleton (which is embedded on the projective plane P). If we now “cut” the plane P along a simple twisted circuit C (an edge of C becoming two edges: one either side of the cut), then we obtain a plane graph H which is bipartite and whose face circuit bounding the infinite face has length 2k for some odd integer k ( k is the length of the circuit C) . We call H a base graph .supporting G . H is not necessarily unique, but does determine G. G can be recovered from H by identifying opposite vertices, and corresponding edges, on the border circuit B ( H ) of H .

Notice that a simple twisted circuit must exist and have odd length otherwise G could not satisfy condition 1 of the definition of a 4-skeleton.

4-skeletons and their supporting base graphs correspond, in a natural way, to the projective plane and its standard “planar representation.” Base graphs provide a representation of 4-skeletons which will be invaluable throughout, not only because they make explicit the (4-circuit) structure of the graph in question (i.e., they make clear the particular embedding of the graph), but also that they allow the author to draw graphs embedded on the projective plane.

As an example, Figure 1 shows a 4-skeleton with one of its supporting base graphs. This 4-skeleton is the well known Mycielski/Grotzsch graph M5 (see [5,8]) .

Suppose R is an orientation of .a graph G, and W is a walk in G. We may partition the edges of W into two sets which we call forward edges and backward edges respectively: those edges whose orientation (given by R ) agrees with their direction of traverse in W are the forward edges, and those whose directions disagree are the backward edges. We denote the sets of forward edges and backward edges by W i and W,, respectively. We now define the $ow-difference f R ( W ) of W in R to be

’ An important property of the flow-difference is its linearity. That is to say, in any orientation R of a graph G and for any walks U and V in G (in fact, for any two 1-chains):

fR(u + v) = fR(u) $- fR(v) .


FIGURE 1. The Mycielski/Grotzsch graph M 5 and a supporting base graph.

We end this section by stating three results which are to be implemented in subsequent sections. The first of these is a well known Coloring Theorem of Minty [9], which for convenience we present in a slightly unusual form, the second is a recent characterization [ I 11 of functions on circuits which are flow-differences, and the third is little more than a combination of the first two.

Theorem 1 (Minty). R in which each circuit C satisfies

A graph G has a k-coloring ( k 2 2) if and only if it has an orientation

Theorem 2 (Pretzel, Youngs). An integer-valued function f defined on the circuits of a graph G is a flow-difference for some orientation R of G if and only if it satisfies for all circuits C and D the following conditions:

( 1 ) If(C)l 5 ICI;

(3) f ( C + D) = f ( C ) + .f(D). (2) f ( C ) = ICI (mod 2);

Corollary 1 A graph G has a k coloring if and only if there exists an integer-valued function f defined on the circuits of G which satisfies, for all circuits C and D , the following conditions:

(1) If(C)I 4 ( k - 2)k-'ICI; (2) .f(c) = ICI (mod 2); (3) f ( C + D ) = f ( C ) + f ( D )

3. COLORING 4-SKELETONS

It is a simple matter to show that each 4-skeleton is precisely 4-chromatic:

Proposition 1. A 4-skeleton G is 4-chromatic.

Proof. First we show that G cannot be 3-colored. Suppose H is a base graph supporting G and R is an orientation of G generated by a 3-coloring (we use integers to color graphs, an orientation is then generated by directing edges from the lesser of their endpoints to the greater). Then R induces an orientation S of H , which itself must be generated by a 3-coloring of H .


For any circuit C of length 4 in H we must have . f R ( C ) = 0. This follows easily from the fact that S is generated by a 3-coloring. In particular, this implies, using the flow-difference additivity, that f s ( B ( H ) ) = 0. We must also have f s ( B ( H ) ) = f ~ ( 2 K ) for some twisted circuit K of G, i.e., K is the circuit of G corresponding to the pre-image of B ( H ) . Because K has odd length we have f ~ ( 2 K ) # 0, giving a contradiction.

Now we show that G can be 4-colored. Suppose that G is k-chromatic for some natural number k, and that G’ is an edge-critical subgraph of G.

G, and therefore G‘, have naturally defined embeddings on the projective plane. Suppose G’ has f face circuits in this embedding. Then the generalized Euler Inequality gives:

Defining S(G’) to be the minimum degree of a vertex of G’, we also have the following elementary inequality:

Because each face circuit of G in the embedding has length 4, then the length of each face circuit of G’ in its embedding is greater than or equal to 4, giving a further inequality:

Straightforward substitution from the second and third inequalities into the first yields:

This implies, using a well known coloring theorem of Dirac [2], that G’ and thus G are

We now turn to the problem of determining precisely which 4-skeletons are edge-critical. at most 4-chromatic. I

Theorem 3. circuit.

A 4-skeleton G is edge-critical if and only if each circuit of length 4 is a face

Proof. First, we remark that all circuits of even length are base circuits. This is because any twisted circuit can be used to form the border circuit of a supporting base graph. If the twisted circuit were even, then a 2-coloring of this base graph would generate a 2 coloring of the 4-skeleton.

Suppose each 4-circuit of G is a face circuit. Let e be any edge of G, and remove e from

We define a function cp on the face circuits of G’ as follows:

if C is a face circuit of length 4 , if C is the (unique) face circuit of length 6 .

G to obtain a graph G’.

0 2

We now extend this function by additivity to each simple circuit of G’. This is clearly possible because the values of p on the face circuits uniquely determine a value for p on every simple circuit.

If C is a base circuit of G’, then it follows from the definition of such circuits that p(C) must take one of the values 0, +2, and -2. Moreover, if cp(C) # 0, then the only face circuit


of G' that C could be is the one of length 6. In any event, it follows from our hypothesis that if p(C) # 0 then C has length greater than or equal to 6.

Now if C is not a base circuit of G' then it must be a twisted circuit. It follows from the definition of a twisted circuit that Ip(2C)l = 2.

Corollary I then implies that G' is 3-colorable, and so the edge e is a critical edge of G. Since t' was an arbitrary edge, then G must be edge-critical.

Conversely, let us suppose that G contains a circuit C of length 4 which is not a face circuit. C is a base circuit, so if we remove all vertices and edges from the interior of C we must obtain a proper subgraph of C which is itself a (4-chromatic) 4-skeleton (the interior of C is the portion of G bounded by C but not including C). Thus, G cannot possibly be critical. I

We conclude this section with a few examples of known graphs which turn out to be 4-skeletons. In each of our examples we picture only a base graph which supports the 4-skeleton in question. The simple task of constructing the corresponding 4-skeleton is left to the over-zealous reader.

The most basic 4-chromatic graphs, the odd wheels W,, are 4-skeletons. Figure 2 shows a base graph supporting W7.

The 4-chromatic Mycielski/Grotzsch Graphs M , [S,S], are 4-skeletons. The best known of these, Mg, and one of its supporting base graphs, was shown in Figure 1. A graph of similar construction to A45 is the 4-chromatic 4-regular triangle-free graph reported by ChvBtal in [ I ] . This graph is of interest to a conjecture of Grunbaum [6]. Its (unique) critical subgraph is a 4-skeleton, and a base graph supporting this graph is shown in Figure 3.

FIGURE 2. A base graph supporting W7

FIGURE 3. A base graph supporting the Chvatal graph


T. Gallai is responsible for a family of 4-skeletons whose property of being 4-chromatic he has shown to be equivalent to the 2-dimensional case of Sperner’s Simplex Lemma [ 131. Gallai’s results are unpublished, but a full account of them as well as some generalizations have been given by Nielsen and Toft in [lo]. Each of the graphs constructed by Gallai consist of a base graph H having border B ( H ) = u l , . . . , ug, u ~ , along with additional edges ( U I , u3),

(u3, ug), and (ug, u ] ) . A base graph supporting such a “Gallai-type” graph may be formed as follows: we simply add to the base graph H , vertices U I , u2, and u3, andedges ( U I , u ~ ) , (u1, u?) , (u2, u3), ( U Z , 2451, (u3,u5) , and ( ~ 3 , U I ) . This base graph has border UI, U I , u3, u2,u5, u3, UI.

4. SHORT ODD CIRCUITS

In 131 Erdos discusses a function f ( r n , k , n). This is defined to be the largest possible chromatic number of an n-vertex graph all of whose rn-vertex induced subgraphs are at most k-chromatic. The value f ( r n , 2, n ) may be described more simply as the maximum chromatic number of an n-vertex graph all of whose odd circuits have length at least rn + 1. We concern ourselves here with the problem of determining the maximum rn for which f ( r n , 2, n ) > 3.

Gallai 141 constructed 4-chromatic graphs which contain no odd circuits of length less than I&]. These graphs immediately imply that

Erdos con.jectured that for a sufficiently large constant c,

In particular he asked if Gallai’s bound is best possible. In [7], Kierstead, Szemerkdi and Trotter proved Erdos’s conjecture to be correct, in fact they

We now construct 4-chromatic graphs on n vertices whose shortest odd circuits have length show that rn is at most S&.

I&] thus improving Gaillai’s bound to give

f ( L & ] , 2, i z ) > 3 .

For each natural number n we describe a graph H,, as follows:

i

j j

V ( H , , ) = {u;: i = 0,1,. . . , n ; J = 1,2, . . ., 2n + I} U { w }

E ( H , , ) = { (u i ,u ;+ l ) : i = 0,1, . . . , n - I ; j = 1 ,2 , . . . , 2n + I} {( ;, ( j + l ) m o d 2 n + l

u’ ui+l ): i = 0, 1 , . . . , n - 1 ; j = I , 2 , . . . , 2n + I }

u { ( w , u i ) : j = 1 , 2 , . . . , 2n + I}.

The H , , ’ s may be viewed as base graphs having border circuit

and so each one supports a 4-skeleton G,. It is easy to see, using Theorem 3, that each G,, is edge-critical. Moreover, G, has 2n2 + n + 1 vertices, and contains no odd circuits with fewer than 2n + 1 edges. The claimed inequality then follows immediately.


The graphs G I and G2 are well known: GI is the complete graph on 4 vertices and Gz is the Mycielski/Grotzsch graph Mg. The graph G1 is shown in Figure 4.

5. FURTHER DEVELOPMENTS

The basic result that 4-skeletons are 4-chromatic is dependent solely on the facts that each face circuit has length 4, the graph is not bipartite, and that the embedding surface is non- orientable. Generalizations concerning the form of surface that the graph must be embedded on appear in [ 121.

An alternative direction is to investigate the particular property of the 4-circuit that makes it so useful: this is that in any 3-coloring of a 4-circuit, the induced orientation on the edges of the circuit is such that the flow-difference of the circuit is zero. Thus, we could “build’ 4-chromatic graphs from copies of any graph which contains a circuit having this “zero flow- difference” property. It would be particularly desirable to have such graphs with an arbitrarily large girth. Little progress has so far been made in this direction, the best attempt known to the author being the graph of Figure 5 which has a girth of 5 , and for which, the bounding 8-circuit has the desired zero flow-difference property.

FlGURE4. The graph G3

FIGURE 5 . A graph having a desirable circuit


As well as increasing the girth as discussed above, it would be equally worthwhile to increase the chromatic number. A possible step in this direction is to jump from a (2-dimensional) projective plane to a higher dimensional projective space. This may not be a fruitful path to follow, and the only evidence the author can suggest in its favor is that the 5-chromatic Mycielski graphs embed pleasantly in projective 3-space in a similar fashion to their 4-chromatic counterparts in 2-space.

Finally, one further possibility for future development would be to characterize the 4-chromatic, triangle-free graphs which are embeddable on the projective plane. Towards this goal we conjecture the following.

Conjecture 1. A triangle-free graph which embeds on the projective plane is 4-chromatic if and only if it contains a 4-skeleton as a subgraph.

A step in the direction of this conjecture has recently been taken by Thomassen [14] who proved that a graph in the projective plane without circuits of length 3 and 4 is 3-colorable.

References

[ I ] V. Chvhtal, The smallest triangle-free 4-chromatic 4-regular graph, J . Comb. Theov 9

[2] G. A. Dirac, A property of 4-chromatic graphs and rome remarks on critical graphs, J. London Math. Soc. 27 (1952), 85-92.

[3] P. Erdiis, Some new application? of probability methods to combinatorical analysis and graph theory, Proc. 5th S.E. Con5 in Comhinatorics, Graph Theory and Computing (1974),

(1970), 93-94.

39-5 1 . 14) T. Gallai, Kritische Graphen, Publ. Math. Inst. Hung. Acad. Sci. 8 (1963), 165-192. [5J H. Grotzsch, Ein Dreifarbensatz fur dreikreisfreie Netze auf der Kugel, Wiw. 2. Murtin-

[6] B. Grunbaum, A problem in graph colouring, Amer. Math. Monthly 77 (1970),

[7] H. A. Kierstead, E. Szemerkdi, and W. T. Trotter, On colouring graphs with locally small

[81 J. Mycielski, Sur les Coloriage des Graphs, Colloq. Math. 3 (1955), 161-162. [9] G. Minty, A theorem on n-colouring the points of a linear graph, Amer. Math. Monthly

[lo] F. Nielsen and B. Toft, On a class of planar 4-chromatic graphs due to T. Gallai, Recent Adv. in Graph Theory, Proc. Symp. Prague 74 Academia Praha, (1975), 425-430.

[ 1 11 0. R. L. Pretzel and D. A. Youngs, Cycle lengths and graph orientations, SIAM J. Discrete

[ 121 0. R. L. Pretzel and D. A. Youngs, Balanced graphs and non-covering graphs, Discrete

[ 131 E. Sperner, Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes, Abh.

[I41 C. Thomassen, Grotzsch’s 3-Color Theorem and Its Counterparts or the Torus and the

[ 151 W. T. Tutte, Graph theory, In: Encyclopedia of Mathematics and its Applications, CUP,

Luther-Univ. Halle-Wittenberg. Math.-Nut. Reihe 8 (1958), 109- 1 19.

1088- 1092.

chromatic number, Combinatorica 4 (1984), 183- 185.

69 ( 1 962), 623 -624.

Math. 3 (4), (1990), 544-553.

Math. 88 (1991), 279-287.

Math. Sem. Hamburg 6 (1928), 265-272.

Projective Plane, J. Comb. Theory, Series B. 62 (1994). 268-279.

Cambridge, (1984), 185-220.

Received February 2, 1995

4-chromatic projective graphs

Documents