Primož Šparl

A classification of tightly attached half-arc-transitive graphs of valency 4

A graph is said to be half-arc-transitive if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so-called alternating cycles is associated, all of which have the same even length. Half of this length is called the radius of the graph in question. Moreover, any two adjacent alternating cycles have the same number of common vertices. If this number, the so-called attachment number, coincides with the radius, we say that the graph is tightly attached. In [D. Marusic, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998) 41-76], Marusic gave a classification of tightly attached half-arc-transitive graphs of valency 4 with odd radius. In this paper the even radius tightly attached graphs of valency 4 are classified, thus completing the classification of all tightly attached half-arc-transitive graphs of valency 4.

On the classification of quartic half-arc-transitive metacirculants

A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms @r and @s, where @r is (m,n)-semiregular for some integers m>=1 and n>=2, and where @s normalizes @r, cyclically permuting the orbits of @r in such a way that @s^m has at least one fixed vertex. In a recent paper Marusic and the author showed that each connected quartic half-arc-transitive metacirculant belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph relative to the semiregular automorphism @r. One of these classes coincides with the class of the so-called tightly-attached graphs, which have already been completely classified. In this paper a complete classification of the second of these classes, that is the class of quartic half-arc-transitive metacirculants for which the quotient graph relative to the semiregular automorphism @r is a cycle with a loop at each vertex, is given.

Some recent discoveries about half-arc-transitive graphs

We present some new discoveries about graphs that are half-arc-transitive (that is, vertex- and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser D 4 (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser of order 16 that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.

Strongly distance-balanced graphs and graph products

A graph G is strongly distance-balanced if for every edge uv of G and every i>=0 the number of vertices x with d(x,u)=d(x,v)-1=i equals the number of vertices y with d(y,v)=d(y,u)-1=i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distance-balanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given.

On strongly regular bicirculants

An n-bicirculant is a graph having an automorphism with two orbits of length n and no other orbits. This article deals with strongly regular bicirculants. It is known that for a nontrivial strongly regular n-bicirculant, n odd, there exists a positive integer m such that n=2m^2+2m+1. Only three nontrivial examples have been known previously, namely, for m=1,2 and 4. Case m=1 gives rise to the Petersen graph and its complement, while the graphs arising from cases m=2 and m=4 are associated with certain Steiner systems. Similarly, if n is even, then n=2m^2 for some m>=2. Apart from a pair of complementary strongly regular 8-bicirculants, no other example seems to be known. A necessary condition for the existence of a strongly regular vertex-transitive p-bicirculant, p a prime, is obtained here. In addition, three new strongly regular bicirculants having 50, 82 and 122 vertices corresponding, respectively, to m=3,4 and 5 above, are presented. These graphs are not associated with any Steiner system, and together with their complements form the first known pairs of complementary strongly regular bicirculants which are vertex-transitive but not edge-transitive.

Hamiltonian cycles in Cayley graphs whose order has few prime factors

We prove that if Cay( G ; S ) is a connected Cayley graph with n vertices, and the prime factorization of n is very small, then Cay( G ; S ) has a hamiltonian cycle. More precisely, if p , q , and r are distinct primes, then n can be of the form kp with 24 ≠ k < 32, or of the form kpq with k ≤ 5, or of the form pqr , or of the form kp 2 with k ≤ 4, or of the form kp 3 with k ≤ 2.

Classification of half-arc-transitive graphs of order 4p

Abstract A vertex-transitive graph X is said to be half-arc-transitive if its automorphism group acts transitively on the set of edges of X but does not act transitively on the set of arcs of X . A classification of half-arc-transitive graphs on 4 p vertices, where p is a prime, is given. Apart from an obvious infinite family of metacirculants, which exist for p ≡ 1 ( mod 4 ) and have been known before, there is an additional somewhat unique family of half-arc-transitive graphs of order 4 p and valency 12 ; the latter exists only when p ≡ 1 ( mod 6 ) is of the form 2 2 k + 2 k + 1 , k > 1 .

Symmetry structure of bicirculants

An n-bicirculant is a graph having an automorphism with two orbits of length n and no other orbits. Symmetry properties of p-bicirculants, p a prime, are extensively studied. In particular, the actions of their automorphism groups are described in detail in terms of certain algebraic representation of such graphs.

Strongly regular tri-Cayley graphs

A graph is called tri-Cayley if it admits a semiregular subgroup of automorphisms having three orbits of equal length. In this paper, the structure of strongly regular tri-Cayley graphs is investigated. A structural description of strongly regular tri-Cayley graphs of cyclic groups is given.

Classification of quartic half-arc-transitive weak metacirculants of girth at most 4

A graph is said to be half-arc-transitive if its automorphism group acts transitively on its vertex set and its edge set, but not on its arc set. A graph is a weak metacirculant if it admits a transitive subgroup of automorphisms { ? , ? } , where ? is semiregular and ? normalizes ? .The main result of this paper is a complete classification of quartic half-arc-transitive weak metacirculants of girth at most 4.