Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and how excluding one fixed cycle length affects the number of colours required as . For vertex-colouring and , if any two distinct vertices connected by a path of at most edges are required to be coloured differently, then a reduction by a logarithmic (in ) factor against the trivial bound can be obtained by excluding an odd cycle length if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by a path of fewer than edges are required to be coloured differently, then excluding an even cycle length is sufficient for a logarithmic factor reduction. For , neither of the above statements are possible for other parity combinations of and . These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).
For a positive integer , the -th power of a (simple) graph is a graph with vertex set in which two distinct elements of are adjacent in if there is a path in of length at most between them. The line graph of a graph is a graph with vertex set in which two distinct elements are adjacent in if the corresponding edges of have a common endpoint. The distance- chromatic number , respectively, distance- chromatic index , of is the chromatic number of , respectively, of . (So is the chromatic number of , the chromatic index of , and the strong chromatic index of .)
The goal of this work is to address the following basic question. What is the largest possible value of or of among all graphs with maximum degree at most that do not contain the cycle of length as a subgraph? For both parameters, we are interested in finding those choices of (depending on ) for which there is an upper bound that is as . (Trivially and are since the maximum degrees and are as . Moreover, by probabilistic constructions [2, 9], these upper bounds must be as regardless of the choice of .) We first discuss some previous work.
For and , the question for essentially was a long-standing problem of Vizing , one that provoked much work on the chromatic number of triangle-free graphs, and was eventually settled asymptotically by Johansson . He used nibble methods to show that the largest chromatic number over all triangle-free graphs of maximum degree at most is as . It was observed in  that this last statement with -free, , rather than triangle-free also holds, thus completely settling this question asymptotically for .
Regarding the question for , first notice that since the chromatic index of a graph of maximum degree is either or , there is little else to say asymptotically if .
For and , the question for was considered by Mahdian  who showed that the largest strong chromatic chromatic index over all -free graphs of maximum degree at most is as . Vu  extended this to hold for any fixed bipartite graph instead of , which in particular implies the statement for any , even. Since the complete bipartite graph satisfies , the statement does not hold for , odd. This completely settles the second question asymptotically for .
In this paper, we advance a systematic treatment of our basic question. Our main results are as follows, which may be considered as extensions of the results of Johansson  and Mahdian  to distance- vertex- and edge-colouring, respectively, for all .
Let be a positive integer and an even positive integer.
For , the supremum of the distance- chromatic number over -free graphs of maximum degree at most is as .
For and , the supremum of the distance- chromatic index over -free graphs of maximum degree at most is as .
Let and be odd positive integers such that . The supremum of the distance- chromatic number over -free graphs of maximum degree at most is as .
Let and be positive integers.
For even and odd, the supremum of the distance- chromatic number over -free graphs of maximum degree at most is as .
For and odd, the supremum of the distance- chromatic index over -free graphs of maximum degree at most is as .
We have reason to suspect that the values and , respectively, may not be improved to lower values in Theorem 1, but we do not go so far yet as to conjecture this. We also wonder whether the value in Theorem 2 is optimal — it might well only be a coincidence for — but we know that in general it may not be lower than , as we show in Section 2.
Our basic question in fact constitutes refined versions of problems of Alon and Mohar  and of Kaiser and the first author , which instead asked about the asymptotically extremal distance- chromatic number and index, respectively, over graphs of maximum degree and girth at least as . Our upper bounds imply bounds given earlier in [2, 9, 10], and the lower bound constructions given there are naturally relevant here (as we shall see in Section 2).
It is worth pointing out that the basic question unrestricted, i.e. asking for the extremal value of the distance- chromatic number or index over graphs of maximum degree as , is likely to be very difficult if we ask for the precise (asymptotic) multiplicative constant. This is because the question for then amounts to a slightly weaker version of a well-known conjecture of Bollobás on the degree–diameter problem , while the question for then includes the notorious strong edge-colouring conjecture of Erdős and Nešetřil, cf. , as a special case.
Our proofs of Theorems 1 and 2 rely on direct applications of the following result of Alon, Krivelevich and Sudakov , which bounds the chromatic number of a graph with bounded neighbourhood density.
Lemma 4 ().
For all graphs with maximum degree at most such that for each there are at most edges spanning , it holds that as .
The proof of this result in  invoked Johannson’s result for triangle-free graphs; using nibble methods directly instead, Vu  extended it to hold for list colouring. So Theorems 1 and 2 also hold with list versions of and .
Section 3 is devoted to showing the requisite density properties for Lemma 4. In order to do so with respect to Theorem 1, we in part use a classic result of Bondy and Simonovits  that the Turán number of the even cycle , that is, the maximum number of edges in a graph on vertices not containing as a subgraph, satisfies as . We also use a technical refinement which we describe and prove in Section 3.
First we review constructions we used in previous work . In combination with the trivial bound if , the following two propositions imply Proposition 3(i). The next result also shows that the value in Theorem 2 may not be reduced below .
Fix . For every even , there exists a -regular graph with and . Moreover, is bipartite if is even, and does not contain any odd cycle of length less than if is odd.
We define as follows. The vertex set is where each is a copy of , the set of ordered -tuples of symbols from . For all , we join an element of and an element of by an edge if the -tuples agree on all symbols except possibly at coordinate , i.e. if for all (and , are arbitrary from ).
It is easy to see that each is a clique in , and every set of edges incident to some is a clique in . This gives and . (In fact here it is easy to find a colouring achieving equality in both cases.)
Since is composed only of bipartite graphs arranged in sequence around a cycle of length , every odd cycle in is of length at least , and is bipartite if is even. ∎
Let be one more than a prime power.
There exists a bipartite, girth , -regular graph with and .
There exists a bipartite, girth , -regular graph with .
There exists a bipartite, girth , -regular graph with .
Letting be the point-line incidence graph of the projective plane , that of a symplectic quadrangle with parameters , and that of a split Cayley hexagon with parameters , it is straightforward to check that these graphs satisfy the promised properties. ∎
In , we somehow combined Propositions 5 and 6 for other lower bound constructions having prescribed girth. This approach is built upon generalised -gons, structures which are known not to exist for . We refer the reader to  for further details.
Our second objective in this section is to introduce a different graph product applicable only to two regular balanced bipartite graphs. We use it to produce two bipartite constructions for , both of which settle the case of even left open in Proposition 5, and the second of which treats what could be interpreted as an edge version of the degree–diameter problem.
Let and be two balanced bipartite graphs with given vertex orderings, i.e. , , , for some positive integers , . We define the balanced bipartite product of and as the graph with vertex and edge sets defined as follows:
See Figure 1 for an example of this product.
Usually the given vertex orderings will be of either of the following types. We say that a labelling , of is a matching ordering of if for all . We say it is a comatching ordering if for all . Note by Hall’s theorem that every non-empty regular balanced bipartite graph admits a matching ordering, while every non-complete one admits a comatching ordering.
Let us now give some properties of this product relevant to our problem, especially concerning its degree and distance properties. The first of these propositions follow easily from the definition.
Let and be two balanced bipartite graphs that have part sizes and , respectively, and are regular of degrees and , respectively, for some positive integers . Suppose , are given in either matching or comatching ordering. Then is a regular balanced bipartite graph with parts and each of size . If both are in matching ordering, then has degree , otherwise it has degree .
Let and be two regular balanced bipartite graphs.
Suppose that for every there is a -path between and in (for some even). Suppose that for every there is a -path between and in (for some even). Then for every , there is a -path between and in .
Suppose that for every there is a -path between and in (for some even). Suppose that for every and there is a -path between and in (for some odd). Then for every and where , there is a -path between and in .
We only show part (ii), as the other part is established in the same manner. Let and . Using the distance assumption on , let be a -path in between and , where is such that . Using the distance assumption on , let be a -path in between and . The following -path between and in traverses using one of the coordinates, then the other:
We use this product to show that no version of Theorem 2 may hold for . In combination with the trivial bound if , we deduce Proposition 3(ii) from Proposition 5, the following result and the fact that .
Fix even. For every with , there exists a -regular bipartite graph with .
Let and . Let be the construction promised by Proposition 5 for and . Since is even, we can write where and . This is a -regular balanced bipartite graph, and for every there exists a -path between and . Moreover, it is possible to label and in comatching ordering so that the indices for coincide with those for for every .
Let and . Let . This is a -regular balanced bipartite graph, and for every , there exists a -path between and . Trivially any labelling of and gives rise to a matching ordering.
Alternatively, Proposition 3(ii) follows from the following result, albeit at the expense of a worse dependency on in the multiplicative factor. For , we can take a -th power of the product operation to produce a bipartite graph of maximum degree with edges such that is a clique.
Fix . For every with , there exists a -regular bipartite graph with and .
Let and , the -th power of under the product , where the factors are always taken in matching ordering. By Proposition 7, is a -regular bipartite graph and has edges. By Proposition 8, there is a path of length at most between every pair of vertices in the same part if is even, or in different parts if is odd. It follows that is a clique. ∎
In this section we prove the main theorems. Before proceeding, let us set notation and make some preliminary remarks.
Let be a graph. We will often need to specify the vertices at some fixed distance from a vertex or an edge of . Let be a non-negative integer. If , we write for the set of vertices at distance exactly from . If , we write for the set of vertices at distance exactly from an endpoint of . We shall often abuse this notation by writing for and so forth. We will write to be the bipartite subgraph induced by the sets and
We will also often need to specify a unique breadth-first search subgraph (, respectively) rooted at (, respectively). Having fixed an ordering of beforehand, i.e. writing , is a graph on whose edges are defined as follows. For every , , we include the edge to that neighbour of in being least in the ordering.
In proving the distance- chromatic number upper bounds in Theorems 1 and 2 using Lemma 4, given , we need to consider the number of pairs of distinct vertices in that are connected by a path of length at most . It will suffice to prove that this number is as for some fixed . In fact, in our enumeration we may restrict our attention to paths of length exactly whose endpoints are in . This is because for all and the number of paths of length exactly containing some fixed vertex is at most for all .
Similarly, in proving the distance- chromatic index upper bound in Theorem 1 using Lemma 4, given , we need to consider the number of pairs of distinct edges that each have at least one endpoint in and that are connected by a path of length at most . It will suffice to prove that this number is as for some fixed . Similarly as above, in our enumeration we may restrict our attention to paths of length exactly whose endpoint edges both intersect .
As mentioned in the introduction, for Theorem 1 we will show a technical refinement of the classic bound on of Bondy and Simonovits . We borrow heavily from the strategy used by Pikhurko  to obtain the following improvement on the bound of Bondy and Simonovits: for all and ,
This has since been improved by Bukh and Jiang . We will require a bound like (1) that only counts edges of a certain type in bipartite -free graphs, but depends on the cardinality of only one of the parts. Given a bipartite graph , we call an edge -bunched with respect to if it is incident to a vertex in of degree at least .
Fix . For any , if is a bipartite -free graph with , then the number of edges that are -bunched with respect to is at most , where .
For a contradiction, let us assume for some that there exists a -free bipartite graph with such that the number of -bunched edges with respect to is more than . Let be the bipartite subgraph of induced by those -bunched edges so that and have smallest cardinality.
Every vertex in has degree at least , and the vertices in have degree at least by assumption. So the average degree of is more than . This implies that contains a subgraph of minimum degree at least where , and so that and .
Let if is odd and otherwise. For every , we define to be the set of vertices at distance from in , and to be the bipartite subgraph of induced by the sets and . We use two intermediary results from  concerning the presence of a -subgraph, defined to be any subgraph that is a cycle of length at least with a chord:
Lemma 12 ().
Let . Any bipartite graph of minimum degree at least contains a -subgraph.
Claim 13 ().
For , contains no -subgraph.
These two statements, together with the fact that there is always a subgraph whose minimum degree is at least half the average degree of the (super)graph, imply that for and , the average degree of must satisfy . Note that since is bipartite there are no edges of spanning for any .
We now show by induction for every that the average degree from to is at most where , i.e. that the number of edges in satisfies
The base case is clearly true since each vertex of has exactly one edge to . Now for the induction let and assume that the statement is true for . By the inductive hypothesis and the properties of , we have
This shows that the average degree from to is at least (where we used that and ). In particular, . It cannot be that the average degree from to is greater than , or else we would not have . So . Combining this with (3), we obtain
Again using the density condition on , this implies
which in turn implies (using that and ) that
By the probabilistic construction described in , it suffices to prove only the upper bound in the statement. We may also assume that , since it was already observed in  that for any the chromatic number of any -free graph of maximum degree is .
Let be even, let be a graph of maximum degree at most such that contains no as a subgraph, and let . Let denote the number of pairs of distinct vertices in that are connected by a path of length exactly . Let . As discussed at the beginning of the section, it suffices for the proof to show that where is a constant independent of , by Lemma 4.
Let us count the possibilities for a path of length between two distinct vertices . Setting , we consider three cases.
The penultimate vertex in the path satisfies . By the assumption that , we obtain that . By (1), the number of edges in is at most . So the number of choices for is at most . The number of choices for the rest of the path is at most .
The penultimate vertex satisfies and is -bunched in with respect to . Then Lemma 11 ensures that there are at most choices for . The number of choices for the rest of the path is at most .
The penultimate vertex satisfies and is not -bunched in with respect to . By the definition of -bunched, there are fewer than choices for given , and so at most choices given . There are at most choices for .
Summing over the above cases, the overall number of choices for the path is at most , giving the required bound on . ∎
By the probabilistic construction described in , it suffices to prove only the upper bound in the statement. To that end, let be even, let be a graph of maximum degree at most such that contains no as a subgraph, and let . Let denote the number of pairs of distinct edges in or that are connected by a path of length . Let . As discussed at the beginning of the section, it suffices to show that where is a constant independent of , by Lemma 4.
Let us count the possibilities for a path where