A graph is said to be 1 2-transitive if its automorphism group is ver-tex and edge but not arc-transitive. For each n 11, a 1 2-transitive graph of valency 4 and girth 6, with the automorphism group isomor-phic to A n Z 2 , is given.

In this paper we describe the structure of locally finite groups in which the bounded nilpotency of two-generator subgroups is a transitive relation. We also introduce the notion of (nilpotent of class c)-transitive kernel. Our results generalize several known results related to the groups in which commutativity is a transitive relation. MSC 2000: 20E15, 20D25

We prove bounds on the chromatic number χ of a vertex-transitive graph in terms of its clique number ω and maximum degree ∆. We conjecture that every vertex-transitive graph satisfies χ 6 max{ω, ⌈ 5∆+3 6 ⌉ }, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with ∆ > 13 we prove the Borodin–Kostochka conjecture, i.e., χ 6 max{ω,∆− 1}.

Abstract: An approach to calculate the exact transitive closure of a parameterized and normalized affine integer tuple relation is presented. A relation is normalized when it describes graphs of the chain topology only. A procedure of the normalization is attached. The exact transitive closure calculation is based on resolving a system of recurrence equations being formed from the input and out...

A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The str...

We consider the class of the topologically locally nite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a nite local representation and a special kind of nite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF...

Characterizations of Egalitarian Characterizations of Egalitarian Characterizations of Egalitarian Characterizations of Egalitarian Binary Relations as Transitive Closures Binary Relations as Transitive Closures Binary Relations as Transitive Closures Binary Relations as Transitive Closures-with a special reference to Lo with a special reference to Lo with a special reference to Lo with a speci...

In this paper we construct transitive t-designs from the linear groups L(2, q), q ≤ 23. Thereby we classify t-designs, t ≥ 2, admitting a transitive action of the linear groups L(2, q), q ≤ 23, up to 35 points and obtained numerous transitive designs, for 36 ≤ v ≤ 55. In many cases we proved the existence of t-designs with certain parameter sets. Among others we constructed t-designs with param...

Two problems of Cameron, Praeger, and Wormald [Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica (1993)] are resolved. First, locally finite highly arc-transitive digraphs with universal reachability relation are presented. Second, constructions of two-ended highly arc-transitive digraphs are provided, where each ‘building block’ is a finite bipartite digrap...

Two questions are considered, namely (i) How many colors are needed for a coloring of the n-cube without monochromatic quadrangles or hexagons? We show that four colors suffice and thereby settle a problem of Erdos. (ii) Which vertex-transitive induced subgraphs does a hypercube have? An interesting graph has come up in this context: If we delete a Hamming code from the 7-cube, the resulting gr...