Jozef Širáň

A Note on Large Graphs of Diameter Two and Given Maximum Degree

Letvt(d,2) be the largest order of a vertex-transitive graph of degreedand diameter 2. It is known thatvt(d,2)=d2+1 ford=1,2,3, and 7; for the remaining values ofdwe havevt(d,2)?d2?1. The only knowngenerallower bound onvt(d,2), valid forall d, seems to bevt(d,2)??(d+2)/2? ?(d+2)/2?. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows thatvt(d,2)?(8/9)(d+12)2for alldof the formd=(3q?1)/2, whereqis a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, ford=7 we obtain as a special case the Hoffman?Singleton graph, and ford=11 andd=13 we have new largest graphs of diameter 2, and degreedon 98 and 162 vertices, respectively.

Exponential Families of Non-Isomorphic Triangulations of Complete Graphs

We prove that the number of non-isomorphic face 2-colourable triangulations of the complete graph Kn in an orientable surface is at least 2n2/54?O(n) for n congruent to 7 or 19 modulo 36, and is at least 22n2/81?O(n) for n congruent to 19 or 55 modulo 108.

Face 2-Colourable Triangular Embeddings of Complete Graphs

A face 2-colourable triangulation of an orientable surface by a complete graphKnexists if and only ifn?3 or 7 (mod12). The existence of such triangulations follows from current graph constructions used in the proof of the Heawood conjecture. In this paper we give an alternative construction for half of the residue classn?7 (mod?12) which lifts a face 2-colourable triangulation byKmto one byK3m?2. A nonorientable version of this result is discussed as well which enables us to produce nonisomorphic nonorientable triangular embeddings ofKnfor half of the residue classn?1 (mod6). We also note the existence of nonisomorphic orientable triangular embeddings ofKnforn?7 (mod12) andn?7.

Approaching the Moore bound for diameter two by Cayley graphs

The order of a graph of maximum degree d and diameter 2 cannot exceed d^2+1, the Moore bound for diameter two. A combination of known results guarantees the existence of regular graphs of degree d, diameter 2, and order at least d^2-2d^1^.^5^2^5 for all sufficiently large d, asymptotically approaching the Moore bound. The corresponding graphs, however, tend to have a fairly small or trivial automorphism group and the nature of their construction does not appear to allow for modifications that would result in a higher level of symmetry. The best currently available construction of vertex-transitive graphs of diameter 2 and preassigned degree gives order 89(d+12)^2 for all degrees of the form d=(3q-1)/2 for prime powers q=1mod4. In this note we show that for an infinite set of degrees d there exist Cayley, and hence vertex-transitive, graphs of degree d, diameter 2, and order d^2-O(d^3^/^2).

The genera, reflexibility and simplicity of regular maps

This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g −1, where g is the genus, all orientably-regular maps of genus p+1 for p prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Siraň classification of non-orientable regular maps of Euler characteristic −p where p is prime.

Regular hypermaps over projective linear groups

An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2,q) or PGL(2,q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.

A note on the number of graceful labellings of paths

With the help of a simple recursive construction we give a computer-assisted proof that the number of graceful labellings of a path of length n grows asymptotically at least as fast as (5/3)n. Results of this type have found surprising applications in topological graph theory.

Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups

Let C C ( d , 2 ) and A C ( d , 2 ) be the largest order of a Cayley graph of a cyclic and an Abelian group, respectively, of diameter 2 and a given degree d . There is an obvious upper bound of the form C C ( d , 2 ) ? A C ( d , 2 ) ? d 2 / 2 + d + 1 . We prove a number of lower bounds on both quantities for certain infinite sequences of degrees d related to primes and prime powers, the best being C C ( d , 2 ) ? ( 9 / 25 ) ( d + 3 ) ( d - 2 ) and A C ( d , 2 ) ? ( 3 / 8 ) ( d 2 - 4 ) . We also offer a result for Cayley graphs of metacyclic groups for general degree and diameter.

Triangular embeddings of complete graphs from graceful labellings of paths

We show that to each graceful labelling of a path on 2s+1 vertices, s>=2, there corresponds a current assignment on a 3-valent graph which generates at least 2^2^s cyclic oriented triangular embeddings of a complete graph on 12s+7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2s+1 vertices grows asymptotically at least as fast as (5/3)^2^s, this method gives at least 11^s distinct cyclic oriented triangular embedding of a complete graph of order 12s+7 for all sufficiently large s.

A note on large Cayley graphs of diameter two and given degree

For a variety of infinite sets of positive integers d related to odd prime powers we describe a simple construction of Cayley graphs of diameter two and given degree d which have order close to d^2/2.