The degree-diameter problem for sparse graph classes

The degree-diameter problem
for sparse graph classes

Guillermo Pineda-Villavicencio  and  David R. Wood
July 5, 2019
Abstract.

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree and diameter . For fixed , the answer is . 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 , and for graphs of bounded arboricity the answer is , in both cases for fixed . 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.

2000 Mathematics Subject Classification:
Centre for Informatics and Applied Optimisation, Federation University Australia, Ballarat, Australia (work@guillermo.com.au). Research supported by a postdoctoral fellowship funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev, and by an ISF grant. Corresponding author.
School of Mathematical Sciences, Monash University, Melbourne, Australia (david.wood@monash.edu). Research supported by the Australian Research Council.

1. Introduction

Let be the maximum number of vertices in a graph with maximum degree at most and diameter at most . Determining is called the degree-diameter problem and is widely studied, especially motivated by questions in network design; see [22] for a survey. Obviously, is at most the number of vertices at distance at most from a fixed vertex. For (which we implicitly assume), it follows that

This inequality is called the Moore bound. The best lower bound is

for some function . For example, the de Bruijn graph shows that ; see Lemma 1. Canale and Gómez [3] established the best known asymptotic bound of for sufficiently large .

For a class of graphs , let be the maximum number of vertices in a graph in  with maximum degree at most and diameter at most . We consider for some particular classes  of sparse graphs, focusing on the case of small diameter , and large maximum degree . We prove lower and upper bounds on of the form

(1)

for some functions and . Since is assumed to be small compared to , the most important term in such a bound is . Thus our focus is on with 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 is for some function ; 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 , which is a big improvement over the unrestricted bound of . 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 max. number of vertices
general
3-colourable
triangle-free 3-colourable
bipartite
average degree
arboricity
treewidth odd
treewidth even
Euler genus odd
Euler genus even
trees

First consider the class of graphs with average degree . In this case, we prove that the maximum number of vertices is for some function (see Section 3). This shows that by assuming bounded average degree we obtain a modest improvement over the standard bound of . A much more substantial improvement is obtained by considering arboricity.

The arboricity of a graph is the minimum number of spanning forests whose union is . Nash-Williams [24] proved that the arboricity of equals

(2)

where the maximum is taken over all subgraphs of . For example, it follows from Euler’s formula that every planar graph has arboricity at most , and every graph with Euler genus has arboricity at most . More generally, every graph that excludes a fixed minor has bounded arboricity. Note that for every graph with minimum degree , average degree , and arboricity . 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 the maximum number of vertices is for some function (see Section 4). Thus by moving from bounded average degree to bounded arboricity the term discussed above is reduced from to . This result generalises the above-mentioned bound for trees, which have arboricity 1. The dependence on in 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 can be defined to be the minimum integer such that is a spanning subgraph of a chordal***A graph is chordal if every induced cycle is a triangle. graph with no -clique. For example, trees are exactly the connected graphs with treewidth 1. See [26, 2] 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 is within a constant factor of if is odd, and of if is even (and is large). These results immediately imply the best known bounds for graphs of given Euler genusA surface is a non-null compact connected 2-manifold without boundary. Every surface is homeomorphic to the sphere with handles or the sphere with cross-caps. The sphere with handles has Euler genus , and the sphere with cross-caps has Euler genus . The Euler genus of a graph is the minimum Euler genus of a surface in which embeds. See the monograph by Mohar and Thomassen [23] for background on graphs embedded in surfaces., 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 , there are 3-colourable graphs with vertices, and for for , there are triangle-free 3-colourable graphs with vertices. These results are in contrast to the bipartite case, in which is the answer.

All undefined terminology and notation is in reference [9].

2. Basic Constructions

This section gives some graph constructions that will later be used for proving lower bounds on .

Lemma 1.

For all integers and the de Bruijn graph has vertices, maximum degree at most , and diameter . Moreover, for , there are sets of vertices in , each containing or vertices, such that each vertex of is in exactly two of the , and the endpoints of each edge of are in some .

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 -inout-regular if each vertex has indegree and outdegree (where a loop at counts in the indegree and the outdegree of ). A digraph has strong diameter if for all (not necessarily distinct) vertices and there is a directed walk from to of length exactly .

Let be the de Bruijn digraph [4, 16], which has vertices, is -inout-regular, and has diameter . Fiol et al. [14, Sec. IV], and Zhang and Lin [32] showed that the can be constructed recursively as a line digraph, as we now explain. If is a digraph with arc set , then the line digraph has vertex set , where is an arc of for all distinct arcs .

Let be the -vertex digraph in which every arc is present (including loops). Now recursively define . The digraph is -inout-regular and has strong diameter ; see [14, Sec. IV].

Define to be the undirected graph that underlies (ignoring loops, and replacing bidirectional arcs by a single edge). Then has vertices, has maximum degree at most , and has (undirected) diameter (since loops can be ignored in shortest paths).

It remains to prove the final claim of the lemma, where . For each vertex of , let be the set of vertices of that correspond to non-loop arcs incident with in . Thus equals or depending on whether there is a loop at in . Each vertex of corresponding to an arc of is in exactly two of these sets, namely and . The endpoints of each edge of corresponding to a path of are both in . These sets, one for each vertex of , define the desired sets in . ∎

The next two lemmas will be useful later.

Lemma 2.

For every integer there is a -regular graph with vertices, containing cliques each of order , such that each vertex in is in exactly two of the , and for all .

Proof.

Let be the line graph of the complete graph . That is, , where is a clique for each . The claimed properties are immediate. ∎

Lemma 3.

For all integers and and there is a bipartite graph with bipartition , such that consists of vertices each with degree , and consists of vertices each with degree at most , and every pair of vertices in have a common neighbour in .

Proof.

By Lemma 2, there is a set of size , containing subsets each of size , such that each element of is in exactly two of the , and for all .

Let be the graph with vertex set , where is defined as follows. For each add a set of vertices to , each adjacent to every vertex in . Since , each vertex in has degree . Since each element of is in exactly two of the , each vertex in has degree .

Consider two vertices . Say and . Let be a vertex in . Then is a common neighbour of and in .

We have proved that has the desired properties in the case that . Finally, delete vertices from , and the obtained graph has the desired properties. ∎

3. Average Degree

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.

Proposition 4.

Every graph with minimum degree , maximum degree and diameter has at most vertices.

Proof.

Let be a vertex of degree . For , let be the number of vertices at distance from . Thus and for all . In total, . ∎

Since minimum degree is at most average degree, we have the following corollary.

Corollary 5.

Every graph with average degree , maximum degree and diameter has at most vertices.

The following is the main result of this section; it says that Corollary 5 is within a constant factor of optimal for fixed .

Proposition 6.

For all integers and and there is a graph with average degree at most , maximum degree at most , diameter at most , and at least vertices.

Proof.

Let . Let . Let . Note that and .

Let be the graph from Lemma 1 with maximum degree at most , diameter , and vertices.

Let be the -regular graph from Lemma 2 with vertices, containing cliques each of order , such that each vertex in is in exactly two of the , and for all .

Let be the cartesian product graph . Note that has vertices and has maximum degree at most . For and , let be the clique in . Since each vertex in is in exactly two of the , each vertex in is in exactly two of the .

Let be the graph obtained from as follows: for and , add an independent set of vertices to completely adjacent to ; that is, every vertex in is adjacent to every vertex in . We now prove that has the claimed properties.

The number of vertices in is

To determine the diameter of , let and be vertices in . Say and . Let be a vertex in . Let be a path of length at most in . Then is path of length at most in . Hence has diameter at most .

Consider the maximum degree of . Each vertex in some set has degree . Each vertex in some set has degree . Thus has maximum degree at most .

It remains to prove that the average degree of is at most . There are vertices of degree at most , and there are vertices of degree . Thus the average degree is at most

Hence it suffices to prove that . Since and ,

That is, . Since and ,

That is, , as desired. Hence the average degree of is at most . ∎

Note that for particular values of 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.

4. Arboricity

This section proves that the maximum number of vertices in a graph with arboricity is for some function . Reasonably tight lower and upper bounds on are established. First we prove the upper bound.

Theorem 7.

For every graph with arboricity , diameter , and maximum degree ,

Proof.

Let be spanning forests of whose union is . Orient the edges of each component of each towards a root vertex. Thus each vertex of has outdegree at most 1 in each ; therefore has outdegree at most in .

Consider an unordered pair of vertices . Let be a shortest -path in . Say has edges. Then . An edge of oriented in the direction from to is called forward. If at least of the edges in are forward, then charge the pair to , otherwise charge to .

Consider a vertex . If some pair is charged to then there is path of length from to with exactly forward arcs, for some and with . Since each vertex has outdegree at most , the number of such paths is at most . Hence the number of pairs charged to is at most

Hence, the total number of pairs, , is at most . The result follows. ∎

We now show that the upper bound in Theorem 7 is close to being best possible (for fixed ).

Theorem 8.

For all even integers and and , such that or , there is a graph with arboricity at most , maximum degree at most , diameter at most , and at least vertices.

Proof.

Let and and . Then , and are positive integers. Let . Then is a positive integer (since or ).

Let be the de Bruijn graph . By Lemma 1, has diameter and vertices. Moreover, there are sets of vertices in , each containing or vertices, such that each vertex of is in exactly two of the , and the endpoints of each edge in are in some . Let . Thus .

By Lemma 3, for each there is a bipartite graph with bipartition , such that consists of vertices each with degree , and consists of vertices each with degree at most , and each pair of vertices in have a common neighbour in .

Let be the bipartite graph with bipartition , where and the induced subgraph is . In , each vertex in has degree , and each vertex in has degree at most . Thus has maximum degree . Assign each edge in one of colours, such that two edges receive distinct colours whenever they have an endpoint in in common. Each colour class is a star forest. Hence has arboricity at most . Observe that

It remains to prove that has diameter at most . Consider two vertices and in . If then let be a neighbour of in . If then let be . If then let be a neighbour of in . If then let be . In , there is a -path of length at most . For each edge in , both and are in some set (see Lemma 1). Since and have a common neighbour in (by Lemma 3), we can replace in by a 2-edge path in , to obtain a -path in of length at most . Possibly adding the edges or gives a -path in of length at most . Hence has diameter at most . ∎

Consider the case of diameter graphs with arboricity . Every such graph has average degree less than , and thus has at most 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 .)

Proposition 9.

For all integers and even there is a graph with diameter , arboricity at most , maximum degree , and at least vertices.

Proof.

By Lemma 2, there is a -regular graph with vertices, containing cliques each of order , such that each vertex in is in exactly two of the , and for all .

Initialise a graph equal to . For , add an independent set of vertices to completely adjacent to .

Consider two vertices and in . Say and . Let be the vertex in . If or then is an edge in , otherwise is a path in . Thus has diameter 2.

Vertices in each have degree and vertices in each have degree . Hence has maximum degree . The number of vertices in is more than .

To calculate the arboricity of , consider a subgraph of . Let and . Since and ,

Since ,

Observe that and (since each vertex in is in exactly two of the ). Thus , and has arboricity at most by (2). ∎

We conclude this section with an open problem about the degree-diameter problem for graphs containing no -minor. Every such graph has arboricity at most , for some constant ; see [21, 29, 28]. Thus Theorem 7 implies that for every -minor-free graph with diameter and maximum degree ,

Improving the term in this bound is a challenging open problem.

5. Separators and Treewidth

This section studies a separator-based approach for proving upper bounds in the degree-diameter problem. A separation of order in an -vertex graph is a partition of , such that and and and there is no edge between and . 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 has vertices. The following lemma improves the dependence on in this result when is even. We include the proof by Fellows et al. [12] for completeness.

Lemma 10.

Let be a graph with maximum degree at most , and diameter at most . Assume is a separation of order in . Then

Proof.

Let . Note that . By symmetry, . We use this fact repeatedly.

For , let . If for some and for some , then , which is a contradiction. Hence, without loss of generality, for each . By the Moore bound, for each vertex , there are at most vertices in at distance at most from . Each vertex in is thus counted. Hence

implying . This proves the result of Fellows et al. [12] mentioned above, and proves the case of odd in the theorem.

Now assume that is even. Suppose on the contrary that

First consider the case in which some vertex in is at distance at least from . Thus every vertex in is at distance at most from . By the Moore bound,

which is a contradiction. Now assume that every vertex in is at distance at most from . By symmetry, every vertex in is at distance at most from .

Let and be the subsets of and respectively at distance exactly from . By the Moore bound, . Hence

By symmetry, .

Let . For each pair , some vertex in is at distance from both and . Charge to . We now bound the number of pairs in charged to each vertex . Say has degree in and degree in . Thus . There are at most vertices at distance exactly from in , and there at most vertices at distance exactly from in . Thus the number of pairs charged to is at most

Hence

This contradiction proves that . ∎

Lemma 10 can be written in the following convenient form.

Lemma 11.

For all there is a constant such that for every graph with maximum degree , diameter , and a separation of order ,

Proof.

First consider the the odd case. For we have . Thus, by Lemma 10 and the Moore bound,

Now consider the even case. For we have . And for we have , implying . Hence, by Lemma 10 and the Moore bound,

Treewidth is a key topic when studying separators. In particular, every graph with treewidth has a separation of order , and in fact, a converse result holds [26]. Thus Lemma 11 implies:

Theorem 12.

For all there is a constant such that for every graph with maximum degree , treewidth , and diameter ,

Note that Theorem 12 in the case of odd can also be concluded from a result by Gavoille et al. [15, Theorem 3.2]. Our original contribution is for the even case. We now show that both upper bounds in Theorem 12 are within a constant factor of optimal.

Proposition 13.

For all integers and and there is a graph with maximum degree , diameter , treewidth at most , and

Proof.

First consider the case of odd . Let be the rooted tree such that the root vertex has degree , every non-root non-leaf vertex has degree , and the distance between the root and each leaf equals . Since and ,

Take disjoint copies of , and add a clique on their roots. This graph is chordal with maximum clique size . Thus it has treewidth . The maximum degree is and the number of vertices is at least .

Now consider the case of even . Let be the maximum integer such that . Thus and . Let be the tree, rooted at , such that has degree , every non-leaf non-root vertex has degree , and the distance between and each leaf is . Since and ,

By Lemma 2, there is a -regular graph with vertices, containing cliques each of order , such that each vertex in is in exactly two of the , and for all . Let be the graph obtained from as follows. For each , add disjoint copies of (called -copies), where every vertex in is adjacent to the roots of the -copies of , as illustrated in Figure 1. It is easily verified that has maximum degree . Consider a vertex in some -copy of or in , and a vertex in some -copy of or in . Let be in . Then and