Constructions of Large Graphs on Surfaces
aa r X i v : . [ m a t h . C O ] F e b CONSTRUCTIONS OF LARGE GRAPHS ONSURFACES
RAMIRO FERIA-PUR ´ON AND GUILLERMO PINEDA-VILLAVICENCIO
Abstract.
We consider the degree/diameter problem for graphsembedded in a surface, namely, given a surface Σ and integers ∆and k , determine the maximum order N (∆ , k, Σ) of a graph em-beddable in Σ with maximum degree ∆ and diameter k . We intro-duce a number of constructions which produce many new largestknown planar and toroidal graphs. We record all these graphs inthe available tables of largest known graphs.Given a surface Σ of Euler genus g and an odd diameter k , thecurrent best asymptotic lower bound for N (∆ , k, Σ) is given by r g ∆ ⌊ k/ ⌋ . Our constructions produce new graphs of order ( ⌊ k/ ⌋ if Σ is the Klein bottle (cid:16) + q g + (cid:17) ∆ ⌊ k/ ⌋ otherwise,thus improving the former value by a factor of 4. Introduction
Given a class C of graphs and integers ∆ and k , the degree/diameterproblem aims to find the maximum order N (∆ , k, C ) of a graph in C with maximum degree ∆ and diameter k . For background on thisproblem the reader is referred to the survey [6].Given a surface Σ, let G (Σ) denote the class of graphs embeddablein Σ. We set N (∆ , k, Σ) := N (∆ , k, G (Σ)) for simplicity.The Moore bound1 + ∆ + ∆(∆ −
1) + . . . + ∆(∆ − k − provides an upper bound for the order of an arbitrary graph with max-imum degree ∆ and diameter k . This bound, however, is a very roughupper bound when considering graphs on surfaces. The current bestupper bound for N (∆ , k, Σ) was provided by ˇSiagiov´a and Simanjuntak
Mathematics Subject Classification.
Primary 05C10; Secondary 05C35.
Key words and phrases. degree/diameter problem, graphs on surfaces, MapColouring Theorem.Guillermo would like to thank the partial support received by the AustralianResearch Council Project DP110102011. [11] who showed that, for every surface Σ of Euler genus g , every k ≥ ≥ N (∆ , k, Σ) ≤ cgk ∆ ⌊ k/ ⌋ . Before continuing we clarify what we mean by a surface and its Eulergenus, as different authors adopt different definitions. A surface is acompact (connected) 2-manifold (without boundary). Every surface ishomeomorphic to the sphere with h handles or the sphere with c cross-caps [7, Theorem 3.1.3]. The sphere with h handles has Euler genus h , while the sphere with c cross-caps has Euler genus c .The bound cgk ∆ ⌊ k/ ⌋ still seems to be rough for graphs on surfacesof Euler genus g , as demonstrated in [10]. For the class P of planargraphs the paper [10] recently showed that, for a fixed k , the limitlim ∆ →∞ N (∆ , k, P )∆ ⌊ k/ ⌋ is an absolute constant, independent of ∆ or k .Knor and ˇSir´aˇn [2] proved that, for every surface Σ, there exists aninteger ∆ such that, for all ∆ ≥ ∆ , N (∆ , , Σ) = N (∆ , , P ) = ⌊
32 ∆ ⌋ + 1 . This result motivated Miller and ˇSir´aˇn to pose the following problem[6, pp. 46].
Problem 1.1 ([6, pp. 46]) . Prove or disprove that, for each surface Σ and each diameter k ≥ , there exists a constant ∆ such that, formaximum degree ∆ ≥ ∆ , N (∆ , k, Σ) = N (∆ , k, P ) . Problem 1.1 was answered in the negative by Pineda-Villavicencioand Wood [10], where the authors proved that, for every surface Σ ofEuler genus g , every odd k ≥ ≥ √ g + 2, N (∆ , k, Σ) ≥ r g ∆ ⌊ k/ ⌋ . In Section 5 we construct graphs whose orders improve the abovelower bound for N (∆ , k, Σ) by a factor of 4. We obtain N (∆ , k, Σ) ≥ ( ⌊ k/ ⌋ if Σ is the Klein bottle (cid:16) + q g + (cid:17) ∆ ⌊ k/ ⌋ otherwise.Sections 3 and 4 are devoted to the construction of new largest knownplanar and toroidal graphs for maximum degree 3 ≤ ∆ ≤
10 anddiameter 2 ≤ k ≤
10. In the appendix we provide tables cataloging thelargest known such graphs. In the case of planar graphs the existingtable was updated with the new orders. For toroidal graphs no suchtable existed; only a table recording largest regular graphs was availableat [9]. We also updated accordingly (created in the case of toroidalgraphs [5]) the online table of largest known planar graphs [4].
ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 3
Our constructions extend approaches put forward in [1]; Section 2explains the methodology.2.
Multigraphs and diagrams
We start from the definition of a diagram presented in [1]. A diagram D is a multigraph where edges are labelled in the form α (∆ , β ) ( α and β positive integers); see Figure 1 (a). An edge in a diagram D is called thin if α = β = 1, otherwise it is called thick . Similarly, a vertex in D iscalled thin if all its incident edges are thin, otherwise it is called thick .For thin edges labels are omitted. The unlabelled degree of a vertex v in D is the number of edges incident with v , while the (labelled) degree ofa vertex v in D is the sum of all the α values on the labels of the edgesincident with v . For instance, in Fig. 1 (a) vertex v has unlabelleddegree two and (labelled) degree three. The weight of a walk in D is the sum of all the β values on the labels of the edges of the walk.An edge e in D with an endvertex of unlabelled degree one is called pending , otherwise e is called non-pending . For an integer k ≥ D k ∆ denotes any diagram D with maximum degree ∆ and labels α (∆ , β )satisfying β ≤ k for non-pending edges, and β ≤ ⌊ k/ ⌋ for pendingedges.The depth of a tree rooted at a vertex v is the length of a longestpath from v to the tree leaves. For a positive integer γ , a (∆ , γ ) -tree isa tree of depht γ with its root and leaves having degree 1 and all othervertices having degree ∆. Given a positive integer β , we call a (∆ , β ) -pod the planar graph obtained from two (∆ , ⌊ β/ ⌋ )-trees, identifyingtheir leaves if β is even and matching their leaves if β is odd; seeFigure 1 (b) for an example. The roots of the two trees used in thepod construction are the roots of the pod; the remaining vertices arecalled internal . In a pod, a path linking its roots is called a vein . Thenumber of internal vertices in a (∆ , β )-pod with β ≥ − ( β − / − − β is even,and 2(∆ − ( β − / − − β is odd.From a diagram D k ∆ we define a compound graph G ( D k ∆ ) as follows:a thick non-pending edge e in D k ∆ , labelled by α (∆ , β ), is replaced by α “disjoint” (∆ , β )-pods. By “disjoint” pods we mean pods that onlyshare their roots. The endvertices of e are identified with the rootsof the pods; see Figure 1 (c). If instead e is a thick pending edge in D k ∆ , e is replaced by α “disjoint” (∆ , β )-trees. The endvertex of e withunlabelled degree other than one is identified with the roots of the treesreplacing e . RAMIRO FERIA-PUR ´ON AND GUILLERMO PINEDA-VILLAVICENCIO
The graph G ( F )2(3 , , F A (3 , , v Figure 1.
Next we establish relations between D k ∆ and G ( D k ∆ ); most of themalready appeared in [1]. Proposition 2.1 ([1, Lemma 1]) . The maximum degree of G ( D k ∆ ) isat most ∆ . Lemma 2.2 ([1, Lemma 2]) . Consider a diagram D k ∆ and the graph G ( D k ∆ ) , and suppose that e and e ′ are two thick edges of D k ∆ which lieon a cycle of weight at most k + 1 . Let v and v ′ be vertices in G ( D k ∆ ) of pods corresponding to e and e ′ respectively. Then the distance in G ( D k ∆ ) between v and v ′ is at most k . Proposition 2.3.
Consider a diagram D k ∆ and the graph G ( D k ∆ ) , andsuppose the following conditions hold. (1) Any two thick edges of D k ∆ are contained in a closed walk ofweight at most k + 1 . (2) For any thin vertex v and any thick edge e of D k ∆ , v and e liein closed walk of weight at most k + 1 . (3) There is a path of weight at most k between any two thin verticesof D k ∆ .Then the graph G ( D k ∆ ) has diameter at most k . ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 5
Proof.
If a thick non-pending edge e of D k ∆ has label α (∆ , β ) then thedistance between any two vertices in G ( D k ∆ ) belonging to the podswhich replaced e is at most k [1, pp. 277]. Since β ≤ ⌊ k/ ⌋ , for athick pending edge e labelled by α (∆ , β ), the distance between anytwo vertices in G ( D k ∆ ) belonging to the trees which replaced e is atmost k as well. Condition (1) assures that any two vertices of G ( D k ∆ )belonging to pods replacing distinct thick edges are at distance at most k . Conditions (2) and (3) guarantee that the distance from a thin vertexto any vertex in a pod of G ( D k ∆ ), and to any other thin vertex, is alsoat most k . (cid:3) As we will show in the proofs of Proposition 3.1 and Lemma4.2,Condition (1) can often be relaxed to containment in two closed walksof weight 2 k + 2, rather than just one closed walk of weight at most2 k + 1. 3. Large planar graphs with odd diameter
Figure 2 (a) depicts the diagram C k ∆ (∆ ≥
4) suggested in [1], whichgives rise to the largest known planar graphs of maximum degree ∆ ≥ k ≥
5. The order of G ( C k ∆ ) for odd k ≥ | G ( C k ∆ ) | = ( ⌊ ⌋ −
12) ∆(∆ − k − − − . As noted in [1], the diagram C k ∆ has maximum degree ∆ and readilysatisfies the conditions of Proposition 2.3. Thus, the graph G ( C k ∆ )has maximum degree ∆ and diameter k . A minor modification of C k ∆ produces a diagram Y k ∆ (for odd k ≥
5) with three additional verticesand, in the case of odd ∆, with an extra pending edge; see Figures 2(b) and (c). Clearly, in both cases Y k ∆ has maximum degree ∆. Proposition 3.1.
For odd k ≥ the diameter of the graph G ( Y k ∆ ) is k .Proof. The diagram Y k ∆ does not satisfy Proposition 2.3, as the pairsof edges ( ac, bg ), ( ac, hi ) and ( bg, hi ), and only those pairs, violateCondition (1). It is not difficult to verify that all other pair of edgesmeet the conditions of Proposition 2.3.The edges ac and bg , however, are contained in the two closed walksof weight 2 k + 2, namely abgeica and adgbxca . Let u be a vertex in G ( Y k ∆ ) of a pod replacing ac , and P the vein containing u . Similarly, let u ′ be a vertex in G ( Y k ∆ ) of a pod replacing bg , and P ′ the vein containing u ′ . We observe that, since k is odd, if u and u ′ are at distance k + 1in the closed walk abP ′ geicP a , then they cannot also be at distance k + 1 in the closed walk adgP ′ bxcP a . This alternative to Condition(1) guarantees that the distance between any two vertices in the podsreplacing ac and bg is at most k . A similar argument applies to thepairs of edges ( ac, hi ) and ( bg, hi ); note the symmetry in Y k ∆ . (cid:3) RAMIRO FERIA-PUR ´ON AND GUILLERMO PINEDA-VILLAVICENCIO
Diagram C k ∆ Diagram Y k ∆ (even ∆)(a) (b) l l l l l l l = ⌊ ∆ − ⌋ (∆ , k − l = ⌈ ∆ − ⌉ (∆ , k − l = (∆ − , k − l l l a cd fe h igb x y z Diagram Y k ∆ (odd ∆)(c) l l = 1(∆ , k − ) l l l l l l a cd fe h igb x y zl l l Figure 2.
Diagrams C k ∆ and Y k ∆ .The number of vertices in G ( Y k ∆ ) is | G ( Y k ∆ ) | = ( | G ( C k ∆ ) | + 3 if ∆ is even | G ( C k ∆ ) | + (∆ − k − − − + 3 if ∆ is oddFor odd k ≥ ≥ G ( Y k ∆ ).When k ≥ Y k ∆ . The resulting diagram Z k ∆ is shown in Figure 3. Proposition 3.2.
For odd k ≥ the diameter of the graph G ( Z k ∆ ) is k .Proof. By virtue of Proposition 3.1, we only need to verify that Con-dition (1) of Proposition 2.3 holds for any pair of thick edges of Z k ∆ , in ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 7 l = ⌊ ∆ − ⌋ (∆ , k − l = ⌈ ∆ − ⌉ (∆ , k − l = (∆ − , k − Z k ∆ (even ∆)(a) l = 1(∆ , k − )Diagram Z k ∆ (odd ∆)(b) l l = (∆ − , ⌊ k − ⌋ ) l l l l l l l l l a cd fe h igb x y zl l l l l l a cd fe h igb x y zl l l Figure 3.
Diagram Z k ∆ .which at least one of the three additional pending edges is implicated.This fact can be verified with little effort. (cid:3) For the new diagram Z k ∆ we have | G ( Z k ∆ ) | = | G ( Y k ∆ ) | + 3(∆ −
2) (∆ − ⌊ k − ⌋ − − . For odd k ≥ ≥ G ( Z k ∆ ).The new record orders obtained from G ( Y k ∆ ) and G ( Z k ∆ ) have beenadded to the table of largest known planar graphs [4], and they are alsodisplayed in Table 1 of the appendix.4. Large graphs embedded in the torus
The diagram-based approach explained in the previous section canbe used to produce large graphs embeddable in an arbitrary surface.
Remark . If a diagram D k ∆ is embeddable in a surface Σ then thegraph G ( D k ∆ ) is also embeddable in Σ.In this section we obtain large graphs in the torus. For our construc-tions we will use the diagrams P k ∆ (for ∆ ≥
3) and Q k ∆ (for ∆ ≥ k ), depicted in Figure 4 (a) and (b), respectively. Sincethe Petersen graph embeds in the torus, the diagram P k ∆ , based on RAMIRO FERIA-PUR ´ON AND GUILLERMO PINEDA-VILLAVICENCIO the Petersen graph, also embeds in the torus. Furthermore, P k ∆ readilysatisfies Proposition 2.3. Thus, we have the following. Diagram P k ∆ Diagram Q k ∆ (a) (b) l = (∆ − , k − l = (∆ − , k − l l l l l l l l l l l l Figure 4.
Diagrams P k ∆ and Q k ∆ . Proposition 4.1.
For any k ≥ the diameter of the graph G ( P k ∆ ) is k . The order of the graph G ( P k ∆ ) is | G ( P k ∆ ) | = ( (cid:0) − k − − (cid:1) + 10 if k is even5 (cid:0) ∆(∆ − k − − (cid:1) + 10 if k is oddAn embedding of Q k ∆ in the torus, based on an embedding of K , ispresented in Fig. 5. We use the drawing solution suggested in [3, Sec-tion 2], where the torus is represented by the inner unshaded rectangle.This rectangle is surrounded by a larger, shaded rectangle, containingcopies of the actual vertices and edges of the embedding. This draw-ing solution allows easy visualisation of the faces and adjacency of theembedding.Next we prove that G ( Q k ∆ ) has diameter at most k . Lemma 4.2.
Let D k ∆ be a diagram for odd k ≥ . Let e = xy and e ′ = x ′ y ′ be two thick edges in D k ∆ , labelled by α (∆ , k − and α ′ (∆ , k − ,respectively. Suppose there is a thin edge in D k ∆ joining x and x ′ , andthin edges f = xy and f = x ′ y ′ parallel to e and e ′ , respectively. Thenthe distance in G ( D k ∆ ) between any vertex u in a pod replacing e andany vertex u ′ in a pod replacing e ′ is at most k . ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 9
Diagram Q k ∆ l = (∆ − , k − l l l l l l l Figure 5.
Embedding of Q k ∆ in the torus based on anembedding K . Proof.
We use a similar argument as in the proof of Proposition 3.1.Note that, also in this case, the thick edges e and e ′ are contained in twoclosed walks of weight 2 k +2 (see Figure 6). Let P and P ′ be the veins in G ( D k ∆ ) containing u and u ′ , respectively. Since k is odd, if u and u ′ areat distance k +1 in the closed walk xx ′ P ′ y ′ f ′ x ′ xP yf x , then they cannotalso be at distance k + 1 in the closed walk xx ′ f ′ y ′ P ′ x ′ xf yP x . (cid:3) l l = α (∆ , k − y x y ′ x ′ l ′ l ′ = α ′ (∆ , k − f ′ f Figure 6.
Auxiliary figure for Lemma 4.2.From Lemma 4.2 it immediately follows
Proposition 4.3.
For odd k ≥ the diameter of the graph G ( Q k ∆ ) is k . From Q k ∆ we obtain | G ( Q k ∆ ) | = − − k − − − + 14 if k is even5(∆ − ∆(∆ − k − − − + 14 if k is oddThe orders for toroidal graphs obtained from P k ∆ and Q k ∆ are dis-played in Table 2. Large graphs on surfaces
As mentioned in the introduction, Pineda-Villavicencio and Wood[10] constructed, for every surface Σ of Euler genus g , every odd di-ameter k ≥ ≥ √ g + 2, graphs withorder r g ∆ ⌊ k/ ⌋ . This is the current best lower bound for N (∆ , k, Σ). In the followingwe improve this lower bound on N (∆ , k, Σ) by a factor of 4, obtainingthe following bound. N (∆ , k, Σ) ≥ ( ⌊ k/ ⌋ if Σ is the Klein bottle (cid:16) + q g + (cid:17) ∆ ⌊ k/ ⌋ otherwise.Our construction modifies a complete graph embedded in the surfaceΣ, so we need the Map Colouring Theorem. This theorem was jointlyproved by Heawood, Ringel and Youngs; see [7, Theorems 4.4.5 and8.3.1]. Theorem 5.1 (Map Colouring Theorem) . Let Σ be a surface withEuler genus g and let G be a graph embedded in Σ . Then χ ( G ) ≤ √ g . Furthermore, with the exception of the Klein bottle where χ ( G ) ≤ ,there is a complete graph G embedded in Σ realising the equality. The right-hand side of the inequality of Theorem 5.1 is called the
Heawood number of the surface Σ and is denoted H (Σ). Define the chromatic number χ of a surface Σ as follows: χ (Σ) = ( H (Σ) otherwise.The main result of this section is the following. Theorem 5.2.
For every surface Σ of Euler genus g , and for every ∆ > ⌈ χ (Σ) − ⌉ + 1 and every odd k ≥ , N (∆ , k, Σ) ≥ χ (Σ) (cid:18) ∆ − − l χ (Σ) − m(cid:19) ∆(∆ − k − − − χ (Σ) . Before proving Theorem 5.2 we recall the operation of vertex split-ting.
Splitting a vertex v consists of replacing v by two adjacent vertices v ′ and v ′′ , and of replacing each edge incident with v by an edge incidentwith either v ′ or v ′′ leaving the other end of the edge unchanged. ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 11
Proof of Theorem 5.2.
We construct large graphs based on a generali-sation of the diagram Q k ∆ in Figure 4 (b). For a given g we construct adiagram Q k ∆ embeddable in a surface Σ of Euler genus g such that thegraph G ( Q k ∆ ) has maximum degree ∆ and diameter k .To obtain Q k ∆ we start from the complete graph K χ (Σ) and an em-bedding of K χ (Σ) in Σ. We split every vertex v in K χ (Σ) as follows. Onthe surface Σ we operate inside a neighbourhood B ǫ ( v ) centred at v ,with radius ǫ small enough so that no vertex of K χ (Σ) other than v iscontained in B ǫ ( v ). Take any edge of K χ (Σ) incident with v and de-note it by e , then denote the other edges incident with v clockwise by e , e , . . . , e χ (Σ) − . Split a vertex v and obtain adjacent vertices v ′ and v ′′ so that the vertex v ′ is incident with the edges e , e , . . . , e ⌊ χ (Σ) − ⌋ andthe vertex v ′′ incident with the edges e ⌊ χ (Σ) − ⌋ +1 , e ⌊ χ (Σ) − ⌋ +2 , . . . , e χ (Σ) − .Then a thick edge v ′ v ′′ labelled by (∆ − − ⌈ χ (Σ) − ⌉ )(∆ , k −
1) isadded; see Figure 7 (b). Note that splitting each vertex v of K χ (Σ) and the subsequent addition of one parallel thick edge do not affect theembeddability in Σ as all these operations are carried out inside theneighbourhood B ǫ ( v ).The resulting diagram Q k ∆ has maximum degree ∆, and so does thegraph G ( Q k ∆ ). The embeddability of G ( Q k ∆ ) follows from the embed-dability of Q k ∆ . Note also that every thick edge in Q k ∆ has a parallelthin edge, and any two thick edges in Q k ∆ are joined by a thin edge.Thus, by Lemma 4.2, the diameter of G ( Q k ∆ ) is k . Finally we have | G ( Q k ∆ ) | = χ (Σ) (cid:18) ∆ − − l χ (Σ) − m(cid:19) ∆(∆ − k − − − χ (Σ) . (cid:3) An example of the construction put forward in Theorem 5.2 wasalready depicted in Fig. 4 (b); see also Fig. 5 for an embedding of sucha construction in the torus.When χ (Σ) is even we can think of one improvement. Since thevertices in Q k ∆ ) arising from v ′ have degree ∆ −
1, it is possible to addan extra pending edge v ′ v ′′ labelled by 1(∆ , k − ); see Figure 7 (c). Thiswould increase the order of G ( Q k ∆ ) by another χ (Σ) (∆ − k − − − vertices.Thus, we have the following. Corollary 5.3.
For every surface Σ of Euler genus g and even χ (Σ) ,and for every ∆ > ⌈ χ (Σ) − ⌉ + 1 and every odd k ≥ , N (∆ , k, Σ) ≥ χ (Σ) (cid:18) ∆ − − l χ (Σ) − m(cid:19) ∆(∆ − k − − − χ (Σ) (∆ − k − − − χ (Σ) . e e e e e ⌊ χ ( g ) − ⌋ e ⌊ χ ( g ) − ⌋ +1 e ⌊ χ ( g ) − ⌋ +2 e ⌊ χ ( g ) − ⌋ +3 e ⌊ χ ( g ) − ⌋ +4 e χ ( g ) − (a) (b) l (c) l l l = (∆ − − ⌈ χ ( g ) − ⌉ )(∆ , k − l = 1(∆ , k − ) . . . . . . . . . . . . . . . . . . v v ′ v ′′ v ′ v ′′ v ′′′ e e e e e e e e Neighbourhood B ǫ ( v ) e ⌊ χ ( g ) − ⌋ e ⌊ χ ( g ) − ⌋ +1 e ⌊ χ ( g ) − ⌋ +2 e ⌊ χ ( g ) − ⌋ +3 e ⌊ χ ( g ) − ⌋ +4 e χ ( g ) − e ⌊ χ ( g ) − ⌋ e ⌊ χ ( g ) − ⌋ +1 e ⌊ χ ( g ) − ⌋ +2 e ⌊ χ ( g ) − ⌋ +3 e ⌊ χ ( g ) − ⌋ +4 e χ ( g ) − Figure 7. Conclusions
Our results and those from [10] imply that, for a fixed odd diame-ter k , N (∆ , k, Σ) is asymptotically larger than N (∆ , k, P ). For evendiameter, however, we believe this is not the case; thus, we dare toconjecture the following. Conjecture 6.1.
For each surface Σ and each even diameter k ≥ ,there exists a constant ∆ such that, for maximum degree ∆ ≥ ∆ , N (∆ , k, Σ) and N (∆ , k, P ) are asymptotically equivalent for a fixed k ;that is, lim ∆ →∞ N (∆ , k, Σ) N (∆ , k, P ) = 1 . Knor and ˇSir´aˇn [2] result for diameter 2 supports this conjecture.For odd k we think the actual assymptotic value of N (∆ , k, Σ) is( c + c √ g )∆ ⌊ k/ ⌋ , where c and c are absolute constants. The case of g = 0 was proved in [10]. ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 13
All the graphs constructed in this paper are non-regular. We couldlook at large regular graphs embedded in surfaces as well. This varia-tion of the degree/diameter problem has already attracted some inter-est; see, for instance, [8]. Such direction merits further attention.
References [1] M. Fellows, P. Hell, and K. Seyffarth,
Constructions of large planar networkswith given degree and diameter , Networks (1998), no. 4, 275–281.[2] M. Knor, and J. ˇSir´aˇn, Extremal graphs of diameter two and given maximumdegree, embeddable in a fixed surface , Journal of Graph Theory (1997), 1–8.[3] W. Kocay, D. Neilson, and R. Szypowski, Drawing graphs on the torus , ArsCombinatoria (2001), 259–277.[4] E. Loz, H. P´erez-Ros´es, and G. Pineda-Villavicencio, The degree/diameter problem for planar graphs , http://combinatoricswiki.org/wiki/The_Degree_Diameter_Problem_for_Planar_Graphs ,2008, accessed on 3 Feb 2013.[5] E. Loz, H. P´erez-Ros´es, and G. Pineda-Villavicencio, The degree/diameter problem for toroidal graphs , http://combinatoricswiki.org/wiki/The_Degree_Diameter_Problem_for_Toroidal_Graphs ,2013, accessed on 3 Feb 2013.[6] M. Miller and J. ˇSir´aˇn, Moore graphs and beyond: A survey of the de-gree/diameter problem , The Electronic Journal of Combinatorics
DS14 (2005),1–61, dynamic survey.[7] B. Mohar, and C. Thomassen,
Graphs on surfaces , Johns Hopkins UniversityPress, Baltimore, 2001.[8] J. Preen,
Largest 6-regular toroidal graphs for a given diameter , The Aus-tralasian Journal of Combinatorics (2010), 53–57.[9] J. Preen, The degree/diameter problem for regular toroidal graphs , http://faculty.cbu.ca/jpreen/torvaldiam.html , 2009, accessed on 3 Feb2013.[10] G. Pineda-Villavicencio, and D. R. Wood, On the degree/diameter problem forminor-closed graph classes , preprint, 2013.[11] J. ˇSiagiov´a, and R. Simanjuntak,
A note on a moore bound for graphs embeddedin surfaces , Acta Mathematica Universitatis Comenianae. New Series (2004),115–117.[12] S.A. Tishchenko, Maximum size of a planar graph with given degree and evendiameter , European Journal of Combinatorics (2012), no. 3, 380–396.[13] Y. Yang, J. Lin, and Y. Dai, Largest planar graphs and largest maximal planargraphs of diameter two , Journal of Computational and Applied Mathematics (2002), no. 1–2, 349–358.
Appendix A. Tables of largest known planar andtoroidal graphs
E-mail address : [email protected] School of Electrical Engineering and Computer Science, The Uni-versity of Newcastle
E-mail address : [email protected] k ∆ FHS
18 28 E FHS FHS FHS
FHS YLD
16 27
FHS
44 81
FHS T FHS
FHS YLD FHS E FHS T FHS T FHS T YLD FHS T T T T YLD FHS T T T T
16 328 FHS FHS T T T
11 124
10 977 T
33 613 FHS FHS T T T
18 698
18 433 T
63 194 FHS FHS T T T
12 301
30 315
30 072 T
110 716
Table 1.
Table of largest known planar graphs in Feb-ruary 2013. Bold entries denote optimal graphs. Under-lined entries correspond to the order of our largest knowngraphs; the old value is also recorded in the same cell. A
Y LD acronym denotes a graph found – or proven to beoptimal – by Yang, Lin and Dai [13]. A
F HS acronymdenotes a graph found by Fellows, Hell and Seyffarth [1].A T acronym denotes a graph found by Tishchenko [12].An E acronym denotes a graph found by Geoffrey Exoo. Centre for Informatics and Applied Optimisation, University ofBallarat
ONSTRUCTIONS OF LARGE GRAPHS ON SURFACES 15 k ∆ r r r r r r r P P r r r r P P P P P r r r P P P P P P r r r P p P p P p p P p P p P p P p
16 328 p P p P p P p P
13 720 p
33 613 p P p Q p Q p Q
23 044 p
63 194 p P p Q p Q p
12 301 Q
38 276 p
110 716
Table 2.
Table of largest known toroidal graphs in Feb-ruary 2013. Entries displaying P and Q denote graphsresulting from diagram P k ∆ and Q k ∆ respectively. An r denotes a largest known regular toroidal graph, whereasa pp