The degree-diameter problem for sparse graph classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\floor{k/2}})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs.


I
Let N (∆, k) be the maximum number of vertices in a graph with maximum degree at most ∆ and diameter at most k. Determining N (∆, k) is called the degree-diameter problem and is widely studied, especially motivated by questions in network design; see [22] for a survey. Obviously, N (∆, k) is at most the number of vertices at distance at most k from a fixed vertex. For ∆ 3 (which we implicitly assume), it follows that This inequality is called the Moore bound. The best lower bound is for some function f . For example, the de Bruijn graph shows that N (∆, k) ∆ 2 k ; see Lemma 1. Canale and Gómez [3] established the best known asymptotic bound of N (∆, k) some particular classes G of sparse graphs, focusing on the case of small diameter k, and large maximum degree ∆. We prove lower and upper bounds on N (∆, k, G) of the form (1) f (k) ∆ g (k) for some functions f and g. Since k is assumed to be small compared to ∆, the most important term in such a bound is g(k). Thus our focus is on g(k) with f (k) a secondary concern.
We first state two straightforward examples, namely bipartite graphs and trees. The maximum number of vertices in a bipartite graph with maximum degree ∆ and diameter k is f (k) ∆ k−1 for some function f ; see references [22,Section 2.4.4] and [1,5]. And for trees, it is easily seen that the maximum number of vertices is within a constant factor of (∆ − 1) k/2 , which is a big improvement over the unrestricted bound of ∆ k . Some of the results in this paper can be thought of as generalisations of this observation.
In what follows we initially consider broadly defined classes of sparse graphs, moving progressively towards more specific classes. The following table summarises our current knowledge, where results in this paper are in bold.
graph class diameter k max. number of vertices general f (k) ∆ k 3-colourable k 2 f (k) ∆ k triangle-free 3-colourable k 4 f (k) ∆ k bipartite f (k) ∆ k−1 average degree d f (k) d∆ k−1 arboricity b f (k, b) ∆ k/2 treewidth t odd k ct (∆ − 1) (k−1)/2 treewidth t even k c √ t (∆ − 1) k/2 Euler genus g odd k c(g + 1)k (∆ − 1) (k−1)/2 Euler genus g even k c (g + 1)k (∆ − 1) k/2 trees c ∆ k/2 First consider the class of graphs with average degree d. In this case, we prove that the maximum number of vertices is f (k) d∆ k−1 for some function f (see Section 3). This shows that by assuming bounded average degree we obtain a modest improvement over the standard bound of (∆ − 1) k . A much more substantial improvement is obtained by considering arboricity.
The arboricity of a graph G is the minimum number of spanning forests whose union is G. Nash-Williams [24] proved that the arboricity of G equals where the maximum is taken over all subgraphs H of G. For example, it follows from Euler's formula that every planar graph has arboricity at most 3, and every graph with Euler genus g has arboricity at most O( √ g). More generally, every graph that excludes a fixed minor has bounded arboricity. Note that δ d 2b for every graph with minimum degree δ, average degree d, and arboricity b. Arboricity is a more refined measure than average degree, in the sense that a graph has bounded arboricity if and only if every subgraph has bounded average degree.
We prove that for a graph with arboricity b the maximum number of vertices is f (b, k) ∆ k/2 for some function f (see Section 4). Thus by moving from bounded average degree to bounded arboricity the g(k) term discussed above is reduced from k − 1 to k 2 . This result generalises the above-mentioned bound for trees, which have arboricity 1. The dependence on b in f can be reduced by making more restrictive assumptions about the graph.
For example, treewidth is a parameter that measures how tree-like a given graph is. The treewidth of a graph G can be defined to be the minimum integer t such that G is a spanning subgraph of a chordal * graph with no (t + 2)-clique. For example, trees are exactly the connected graphs with treewidth 1. See [2,26] for background on treewidth. Since the arboricity of a graph is at most its treewidth, bounded treewidth is indeed a more restrictive assumption than bounded arboricity. We prove that the maximum number of vertices in a graph with treewidth t is within a constant factor of t(∆ − 1) (k−1)/2 if k is odd, and of √ t(∆−1) k/2 if k is even (and ∆ is large). These results immediately imply the best known bounds for graphs of given Euler genus † , and new bounds for apex-minor-free graphs. All these results are presented in Section 5.
Our results in Section 6 are of a different nature. There, we describe (non-sparse) graph classes for which the maximum number of vertices is not much different from the unrestricted case. In particular, we prove that for k 2, there are 3-colourable graphs with f (k) ∆ k vertices, and for for k 4, there are triangle-free 3-colourable graphs with f (k) ∆ k vertices. These results are in contrast to the bipartite case, in which f (k) ∆ k−1 is the answer. All undefined terminology and notation is in reference [9]. * A graph is chordal if every induced cycle is a triangle. † A surface is a non-null compact connected 2-manifold without boundary. Every surface is homeomorphic to the sphere with h handles or the sphere with c cross-caps. The sphere with h handles has Euler genus 2h, and the sphere with c cross-caps has Euler genus c. The Euler genus of a graph G is the minimum Euler genus of a surface in which G embeds. See the monograph by Mohar and Thomassen [23] for background on graphs embedded in surfaces.

B C
This section gives some graph constructions that will later be used for proving lower bounds on N (∆, k, G).
Lemma 1. For all integers r 1 and k 1 the de Bruijn graph B(r, k) has r k vertices, maximum degree at most 2r, and diameter k. Moreover, for k 2, there are sets B 1 , . . . , B r k−1 of vertices in B(r, k), each containing 2r − 2 or 2r vertices, such that each vertex of B(r, k) is in exactly two of the B i , and the endpoints of each edge of Proof. In what follows, a digraph is a directed graph possibly with loops and possibly with arcs in opposite directions between two vertices. A digraph is r-inout-regular if each vertex has indegree r and outdegree r (where a loop at v counts in the indegree and the outdegree of v). A digraph has strong diameter k if for all (not necessarily distinct) vertices v and w there is a directed walk from v to w of length exactly k. can be constructed recursively as a line digraph, as we now explain. If G is a digraph with arc set A(G), then the line digraph L(G) has vertex set A(G), where (uv, vw) is an arc of L(G) for all distinct arcs uv, vw ∈ A(G).
Let − → B (r, 1) be the r-vertex digraph in which every arc is present (including loops). Now Define B(r, k) to be the undirected graph that underlies − → B (r, k) (ignoring loops, and replacing bidirectional arcs by a single edge). Then B(r, k) has r k vertices, has maximum degree at most 2r, and has (undirected) diameter k (since loops can be ignored in shortest paths).
It remains to prove the final claim of the lemma, where k 2. For each vertex v of − → B (r, k − 1), let B v be the set of vertices of B(r, k) that correspond to non-loop arcs . Each vertex of B(r, k) corresponding to an arc vw of − → B (r, k − 1) is in exactly two of these sets, namely B v and B w . The endpoints of each edge of B(r, k) corresponding to a path uv, vw of − → B (r, k − 1) are both in B v . These r k−1 sets, one for each vertex of − → B (r, k − 1), define the desired sets in B(r, k).
The next two lemmas will be useful later.

Lemma 2.
For every integer q 1 there is a (2q −2)-regular graph L with q+1 2 vertices, containing cliques L 1 , . . . , L q+1 each of order q, such that each vertex in L is in exactly two of the L i , and L i ∩ L j = ∅ for all i, j ∈ [1, q + 1].
Lemma 3. For all integers p 1 and q 1 and m (q + 1)p there is a bipartite graph T with bipartition C, D, such that C consists of m vertices each with degree q, and D consists of q+1 2 vertices each with degree at most 2p, and every pair of vertices in C have a common neighbour in D.
Proof. By Lemma 2, there is a set D of size q+1 2 , containing subsets D 1 , . . . , D q+1 each of size q, such that each element of D is in exactly two of the D i , and D i ∩ D j = ∅ for all i, j ∈ [1, q + 1].
Let T be the graph with vertex set C ∪ D, where C is defined as follows. For each i ∈ [1, q + 1] add a set C i of p vertices to C, each adjacent to every vertex in D i . Since |D i | = q, each vertex in C has degree q. Since each element of D is in exactly two of the D i , each vertex in D has degree 2p.
Consider two vertices v, w ∈ C. Say v ∈ C i and w ∈ C j . Let x be a vertex in D i ∩ D j . Then x is a common neighbour of v and w in G.
We have proved that T has the desired properties in the case that m = (q + 1)p. Finally, delete (q+1)p−m vertices from C, and the obtained graph has the desired properties.

A D
This section presents bounds on the maximum number of vertices in a graph with given average degree. For fixed diameter, the upper and lower bounds are within a constant factor. We have the following rough upper bound for graphs of given minimum degree.
Proof. Let v be a vertex of degree δ. For 0 i k, let n i be the number of vertices at distance i from v. Thus n 0 = 1 and n i δ(∆ − 1) i−1 for all i 1. In total, Since minimum degree is at most average degree, we have the following corollary.
The following is the main result of this section; it says that Corollary 5 is within a constant factor of optimal for fixed k.
Proposition 6. For all integers d 4 and k 3 and ∆ 2d there is a graph with average degree at most d, maximum degree at most ∆, diameter at most k, and at least Let B := B(r, k − 2) be the graph from Lemma 1 with maximum degree at most 2r, diameter k − 2, and r k−2 vertices.
Let L be the (2q − 2)-regular graph from Lemma 2 with q+1 2 vertices, containing cliques L 1 , . . . , L q+1 each of order q, such that each vertex in L is in exactly two of the L i , and Let H be the cartesian product graph L B. Note that H has q+1 2 r k−2 vertices and has maximum degree at most 2q − 2 + 2r.
Let G be the graph obtained from H as follows: for i ∈ [1, q + 1] and v ∈ V (B), add an independent set Y i,v of p vertices to G completely adjacent to X i,v ; that is, every vertex in Y i,v is adjacent to every vertex in X i,v . We now prove that G has the claimed properties.

The number of vertices in G is
To determine the diameter of G, let α and β be vertices in G.
has maximum degree at most ∆.
It remains to prove that the average degree of G is at most d. There are |V (H)| = q+1 2 r k−2 vertices of degree at most ∆, and there are (q + 1)r k−2 p vertices of degree q.
Thus the average degree is at most Hence it suffices to prove that q∆ + 2pq (q + 2p)d. Since ∆ 2d and d 4q, That is, 4pd + 2qd 2q∆ + 4pq, as desired. Hence the average degree of G is at most d.
Note that for particular values of k and ∆, other graphs can be used instead of the de Bruijn graph in the proof of Proposition 6 to improve the constants in our results; we omit all these details.

A
This section proves that the maximum number of vertices in a graph with arboricity b is f (b, k) · ∆ k/2 for some function f . Reasonably tight lower and upper bounds on f are established. First we prove the upper bound.
Theorem 7. For every graph G with arboricity b, diameter k, and maximum degree ∆, Proof. Let G 1 , . . . , G b be spanning forests of G whose union is G. Orient the edges of each component of each G i towards a root vertex. Thus each vertex v of G has outdegree at most 1 in each G i ; therefore v has outdegree at most b in G.
Consider an unordered pair of vertices {v, w}. Let P be a shortest vw-path in G. Say P has edges. Then k. An edge of P oriented in the direction from v to w is called forward. If at least 2 of the edges in P are forward, then charge the pair {v, w} to v, otherwise charge {v, w} to w.
Consider a vertex v. If some pair {v, w} is charged to v then there is path of length from v to w with exactly i forward arcs, for some i and with 2 i k. Since each vertex has outdegree at most b, the number of such paths is at most Hence, the total number of pairs, n 2 , is at most 2k(2b) k ∆ k/2 n. The result follows.
We now show that the upper bound in Theorem 7 is close to being best possible (for fixed k).
Theorem 8. For all even integers b 2 and k 4 and ∆ b, such that ∆ ≡ 2 (mod 4) or b ≡ 0 (mod 4), there is a graph G with arboricity at most b, maximum degree at most ∆, diameter at most k, and at least 8 b 2 ( b∆ 8 ) k/2 vertices.
Let B be the de Bruijn graph B(r, ). By Lemma 1, B has diameter and r vertices. Moreover, there are sets B 1 , . . . , B r −1 of vertices in B, each containing 2r − 2 or 2r vertices, such that each vertex of B is in exactly two of the B i , and the endpoints of each edge in B are in some B i . Let r i := |B i |. Thus r i 2r = (q + 1)p.
By Lemma 3, for each i ∈ [1, r −1 ] there is a bipartite graph T i with bipartition B i , D i , such that B i consists of r i vertices each with degree q, and D i consists of q+1 2 vertices each with degree at most 2p b, and each pair of vertices in B i have a common neighbour in D i .
has degree 2q ∆, and each vertex in D has degree at most b ∆. Thus G has maximum degree ∆. Assign each edge in G one of b colours, such that two edges receive distinct colours whenever they have an endpoint in D in common. Each colour class is a star forest. Hence G has arboricity at most b. Observe that It remains to prove that G has diameter at most k. Consider two vertices v and w in G. If there is a v w -path P of length at most . For each edge xy in P , both x and y are in some set B a (see Lemma 1). Since x and y have a common neighbour in T a (by Lemma 3), we can replace xy in P by a 2-edge path in T a , to obtain a v w -path in G of length at most 2 .
Possibly adding the edges vv or ww gives a vw-path in G of length at most 2 + 2 = k. Hence G has diameter at most k.
Consider the case of diameter 2 graphs with arboricity b. Every such graph has average degree less than 2b, and thus has at most 4b∆ vertices by Corollary 5. We now show that this upper bound is within a constant factor of optimal. (This result is not covered by Theorem 8 which assumes k 4.) Proposition 9. For all integers b 1 and even ∆ 4b there is a graph with diameter 2, arboricity at most b, maximum degree ∆, and at least b∆ 4 vertices.
Proof. By Lemma 2, there is a (2b − 2)-regular graph X with b+1 2 vertices, containing cliques X 1 , . . . , X b+1 each of order b, such that each vertex in X is in exactly two of the Vertices in each X i have degree 2b − 2 + 2p = ∆ and vertices in each Y i have degree b ∆. Hence G has maximum degree ∆. The number of vertices in G is more than To calculate the arboricity of G, consider a subgraph H of G. Let − 1), and G has arboricity at most b by (2).
We conclude this section with an open problem about the degree-diameter problem for graphs containing no K t -minor. Every such graph has arboricity at most ct √ log t, for some constant c > 0; see [21,28,29]. Thus Theorem 7 implies that for every K t -minor-free graph G with diameter k and maximum degree ∆ t, Improving the f (t, k) term in this f (t, k) ∆ k/2 bound is a challenging open problem.

S T
This section studies a separator-based approach for proving upper bounds in the degreediameter problem. A separation of order s in an n-vertex graph G is a partition (A, S, B) of V (G), such that |A| 2 3 n and |B| 2 3 n and |S| s and there is no edge between A and B. Fellows et al. [12] first used separators to prove upper bounds in the degree-diameter problem. In particular, they implicitly proved that every graph that has a separation of order s has 3s M (∆, k 2 ) vertices. The following lemma improves the dependence on s in this result when k is even. We include the proof by Fellows et al.
Lemma 10. Let G be a graph with maximum degree at most ∆, and diameter at most k. Assume (A, S, B) is a separation of order s in G. Then Proof. Let n := |V (G)|. Note that |A| n − |B| − s n 3 − s. By symmetry, |B| n 3 − s. We use this fact repeatedly.
For v ∈ A∪B, let dist(v, S) := min{dist(v, x) : x ∈ S}. If dist(v, S) k/2 +1 for some v ∈ A and dist(w, S) k/2 + 1 for some w ∈ B, then dist(v, w) 2 k/2 + 2 k + 1, which is a contradiction. Hence, without loss of generality, dist(v, S) k/2 for each v ∈ A. By the Moore bound, for each vertex x ∈ S, there are at most M (∆, k/2 ) − 1 vertices in A at distance at most k/2 from x. Each vertex in A is thus counted. Hence Now assume that k = 2 is even. Suppose on the contrary that First consider the case in which some vertex in A is at distance at least + 1 from S. Thus every vertex in B is at distance at most − 1 from S. By the Moore bound, which is a contradiction. Now assume that every vertex in A is at distance at most from S. By symmetry, every vertex in B is at distance at most from S.
Let A and B be the subsets of A and B respectively at distance exactly from S. By the Moore bound, |A − A | s M (∆, − 1) − s. Hence For each pair (x, y) ∈ P , some vertex v in S is at distance from both x and y. Charge (x, y) to v. We now bound the number of pairs in P charged to each vertex v ∈ S. Say v has degree a in A and degree b in B. Thus a + b ∆. There are at most a(∆ − 1) −1 vertices at distance exactly from v in A, and there at most b(∆ − 1) −1 vertices at distance exactly from v in B. Thus the number of pairs charged to v is at most This contradiction proves that n 3 Lemma 10 can be written in the following convenient form.
Lemma 11. For all > 0 there is a constant c such that for every graph G with maximum degree ∆, diameter k, and a separation of order s, Proof. First consider the the odd k case. For ∆ 6 + 2 we have 3( ∆ ∆−2 ) 3 + . Thus, by Lemma 10 and the Moore bound,

Now consider the even k case. For
. Hence, by Lemma 10 and the Moore bound, Treewidth is a key topic when studying separators. In particular, every graph with treewidth t has a separation of order t + 1, and in fact, a converse result holds [26]. Thus Lemma 11 implies: Theorem 12. For all > 0 there is a constant c such that for every graph G with maximum degree ∆, treewidth t, and diameter k, Note that Theorem 12 in the case of odd k can also be concluded from a result by Gavoille et al. [15,Theorem 3.2]. Our original contribution is for the even k case. We now show that both upper bounds in Theorem 12 are within a constant factor of optimal.
Proposition 13. For all integers k 1 and t 2 and ∆ there is a graph G with maximum degree ∆, diameter k, treewidth at most t, and Proof. First consider the case of odd k. Let T be the rooted tree such that the root vertex has degree ∆ − t, every non-root non-leaf vertex has degree ∆, and the distance between the root and each leaf equals k−1 2 . Since t 2 and ∆−t Take t + 1 disjoint copies of T , and add a clique on their roots. This graph is chordal with maximum clique size t + 1. Thus it has treewidth t. The maximum degree is ∆ and the number of vertices is at least 1 2 (t + 1)(∆ − 1) (k−1)/2 . Now consider the case of even k. Let q be the maximum integer such that q+1 2 t + 1.
Thus 2 q √ 2t ∆ 4 and q + 1 √ t + 1. Let T be the tree, rooted at r, such that r has degree ∆−q, every non-leaf non-root vertex has degree ∆, and the distance between r and each leaf is k 2 − 1. Since q 2 and ∆−q By Lemma 2, there is a (2q − 2)-regular graph L with q+1 2 vertices, containing cliques L 1 , . . . , L q+1 each of order q, such that each vertex in L is in exactly two of the L i , and L i ∩L j = ∅ for all i, j ∈ [1, q+1]. Let G be the graph obtained from L as follows. For each i ∈ [1, q + 1], add ∆ − 2(q − 1) disjoint copies of T (called i-copies), where every vertex in L i is adjacent to the roots of the i-copies of T , as illustrated in Figure 1. It is easily verified that G has maximum degree ∆. Consider a vertex v in some i-copy of T or in L i , and a vertex w in some j-copy of T or in L j . Let x be in L i ∩ L j . Then dist(v, x) k 2 and dist(w, x) k 2 , implying dist(v, w) k. Hence G has diameter at most k. Let G be the super graph of G obtained by adding a clique on V (L). Thus G is chordal with maximum clique size q+1 2 t + 1. Hence G has treewidth at most t. The number of vertices in G is at least (q + 1)(∆ − 2q + 2)|V (T )| √ t + 1 · ∆ 2 · (∆ − 1) k/2−1 .
F . Construction in Proposition 13 for even k. Here ∆ = 4 and k = 8 and t = 2.
We now consider the degree-diameter problem for graphs with given Euler genus. Note that the case of planar graphs has been widely studied [12,13,19,25,30,31]. Šiagiová and Simanjuntak [27] proved that for every graph G with Euler genus g, for some absolute constant c. Eppstein [11] proved that every graph with Euler genus g and diameter k has treewidth at most c(g + 1)k for some absolute constant c, and Dujmovic et al.
[10] proved the explicit bound of (2g + 3)k. Theorem 12 thus implies the upper bound in (3) and improves upon it when k is even: Theorem 14. For all > 0 there is a constant c such that for every graph G with Euler genus g, maximum degree ∆ and diameter k, if k is even and ∆ c (2g + 3)k + 1 .
In our companion paper [25] we further investigate the degree-diameter problem for graphs on surfaces, providing an improved upper bound and a new lower bound.
To obtain an upper bound of the form n f (k) ∆ k/2 using the separator-based ap- As discussed above, for minor-closed classes, Theorem 15 is the strongest possible result that can be obtained using the separator-based method.
6. -C T -F G As mentioned in the introduction, it is well known that the maximum number of vertices in a bipartite graph is f (k) ∆ k−1 . We now show that this bound does not hold for the more general class of 3-colourable graphs. In fact, we construct 3-colourable graphs where the number of vertices is within a constant factor of the Moore bound. First note that Kawai and Shibata [20] proved (building on the work of Harner and Entringer [18]) that for large k log r, the de Bruijn graph B(r, k), which roughly has ∆ 2 k vertices, is 3-colourable. The constructions below have the advantage of not assuming that k is large.
In what follows a pseudograph is an undirected graph possibly with loops. A loop at a vertex v counts for 1 in the degree of v. A pseudograph H is k-good if for all (not necessarily distinct) vertices v and w there is a vw-walk of length exactly k in H. H 1 and H H 2 ) if and only if vw ∈ E(H 1 ) and xy ∈ E(H 2 ). Lemma 16. Let H 1 and H 2 be k-good pseudographs with maximum degree ∆ 1 and ∆ 2 respectively. Then H 1 × H 2 has |V (H 1 )| · |V (H 2 )| vertices, maximum degree ∆ 1 ∆ 2 , and diameter at most k. Moreover, if H 2 is loopless and c-colourable, then H 1 × H 2 is c-colourable.

Given pseudographs
Proof. Clearly H 1 × H 2 has |V (H 1 )| · |V (H 2 )| vertices and maximum degree ∆ 1 ∆ 2 . Let (v, x) and (w, y) be distinct vertices of G. To prove that G has diameter at most k, we has diameter at most k. Finally, colouring each vertex (v, x) of H 1 × H 2 by the colour assigned to x in a c-colouring of H 2 gives a c-colouring of H 1 × H 2 .
Lemma 17. K 3 is k-good for all k 2.
Proof. Let v, w ∈ V (K 3 ) = {0, 1, 2}. If there is a vw-walk of length k − 2, then there is a vw-walk of length k (just repeat one edge twice). Thus the claim follows from the k = 2 and k = 3 cases. Without loss of generality, v = 0. For k = 2, one of 010, 021 and 012 is a vw-walk of length 2. For k = 3, one of 0120, 0121 and 0102 is a vw-walk of length 3.
Lemmas 16 and 17 imply: Lemma 19. Let H be a k-good pseudograph with maximum degree ∆ for some k 2. Then H × K 3 is a 3-colourable graph with 3|V (H)| vertices, maximum degree 2∆, and diameter at most k.
Lemma 20. Let H be a k-good pseudograph with maximum degree ∆ for some k 4. Then H ×C 5 is a 3-colourable triangle-free graph with 5|V (H)| vertices, maximum degree 2∆, and diameter at most k.
Proof. For any graph G (without loops), if H × G contains a triangle (a, u)(b, v)(c, w), then uvw is a triangle in G (even if H has loops). Since C 5 is triangle-free, H × C 5 is triangle-free. Thus Lemmas 16 and 18 imply the claim.
For particular values of ∆ and k, various constructions for the degree-diameter problem can be used in the following lemma to give large 3-colourable and triangle-free graphs.

Proposition 21.
Let H be a graph with maximum degree ∆ and diameter k 2. Then there is a 3-colourable graph with 3|V (H)| vertices, maximum degree 2∆+2 and diameter at most k. Moreover, if k 4 then there is a 3-colourable triangle-free graph with 5|V (H)| vertices, maximum degree 2∆ + 2, and diameter at most k.

Proof. Let H be the pseudograph obtained from H by adding a loop at each vertex. Thus
H is k-good and has maximum degree ∆ + 1. Lemmas 19 and 20 imply that H × K 3 and H × C 5 satisfy the claims.
This result implies that for fixed k 2 and ∆ k, the maximum number of vertices in a 3-colourable graph is within a constant factor of the unrestricted case. And the same conclusion holds for k 4 for 3-colourable triangle-free graphs.
We now give a concrete example: Theorem 22. For all integers ∆ 4 and k 2, there is a 3-colourable graph with 3 ∆ 4 k vertices, maximum degree at most ∆, and diameter at most k. Moreover, if k 4 then there is a 3-colourable triangle-free graph with 5 ∆ 4 k vertices, maximum degree at most ∆, and diameter at most k.
Proof. Let r := ∆ 4 . Let H be the undirected pseudograph underlying the de Bruijn digraph − → B (r, k) including any loops. Lemma 1 shows that H has r k vertices, maximum degree at most 2r, and is k-good. Lemma 19 shows that H × K 3 satisfies the first claim. Lemma 20 implies that H × C 5 satisfies the second claim.
We now give ad-hoc constructions of triangle-free graphs with diameter 2 and 3. These lower bounds are within a constant factor of the Moore bound. Let Z p be the cyclic group with p elements. For a, b ∈ Z p , let dist(a, b) := min{a − b, b − a}. Here, as always, addition is in the group.
Suppose on the contrary that G contains a triangle T . For each edge uv of T , we have dist(u i , v i ) = 1 for some i ∈ [1, 2]. In this case, say uv is type i. Since there are three pairs of vertices in T and only two types, two pairs of vertices in T have the same type. Say T = uvw. Without loss of generality, uv and vw are both type-1. That is, dist(u 1 , v 1 ) = 1 and dist(v 1 , w 1 ) = 1. Thus dist(u 1 , w 1 ) ∈ {0, 2}, in which case uw ∈ E(G). This contradiction shows that G contains no triangle.
Proof. Let p := 2 ∆+6 12 + 4. Thus p 12 is even. Let H be the graph with vertex set Z p , where ab ∈ E(H) whenever dist(a, b) 3. Observe that every pair of vertices in H have a common neighbour (since p 12). if and only if a = 0 and b = 1 and c 3. Observe that G is 6(p − 5)-regular, and 6(p − 5) ∆.
We now show that the distance between distinct vertices v, w in G is at most 3. Consider the following cases for the vw-vector, where without loss of generality, (a, b, c) = (dist(v 1 , w 1 ), dist(v 2 , w 2 ), dist(v 3 , w 3 )): Case (0, 0, 1): Let u be a common neighbour of v 3 and w 3 in H. Then (v 1 + 1, v 2 , u) = (w 1 + 1, w 2 , u) is a common neighbour of v and w.
is a common neighbour of v and w.
Thus G has diameter at most 3.
Suppose on the contrary that G contains a triangle T . For each edge uv of T , we have u i = v i for exactly one value of i ∈ [1, 3]. In this case, say uv is type i. First suppose that at least two of the edges in T are the same type. Then all three edges in T are the same type. Without loss of generality, u 1 = v 1 = w 1 . Then the subgraph of G induced by {u, v, w} (ignoring the first coordinate) is a subgraph of the graph in Proposition 23, which is triangle-free. Now assume that all three edges in T have distinct types. Without loss of generality, u 1 = v 1 and u 2 = w 2 and v 3 = w 3 . Since uv ∈ E(G), without loss of generality, dist(u 2 , v 2 ) = 1 and dist(u 3 , v 3 ) 3. Thus dist(v 2 , w 2 ) = 1 and dist(u 3 , w 3 ) 3. Since vw ∈ E(G) and u 1 = v 1 , we have dist(u 1 , w 1 ) = dist(v 1 , w 1 ) 3.
Finally, note that the graphs in Propositions 23 and 24 have bounded chromatic number. In Proposition 23, colour each vertex v by (v 1 mod 2, v 2 mod 2). For each edge vw, we have dist(v i , w i ) = 1 for some i. Since p is even, v i ≡ w i (mod 2). Thus this is a valid 4colouring. In Proposition 24, colouring each vertex v by (v 1 mod 2, v 2 mod 2, v 3 mod 2) gives an 8-colouring.

A
The case k = 2 of Theorem 12 was proved in collaboration with Bruce Reed. Thanks Bruce.