Almost all almost regular c-partite tournaments with c⩾5 are vertex pancyclic

Almost all almost regular c-partite tournaments with cgeq5 are vertex pancyclic

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + (x) be the outdgree and d ? (x) the indegree of the vertex x in D. The minimum (maximum) out-degree and the minimum (maximum) indegree of D are denoted by + ((+) and ? ((?), respectively. In addition, we deene = minf + ; ?...

Diregular c-partite tournaments are vertex-pancyclic when c ≥ 5

In 4] it is conjectured that all diregular c-partite tournaments, with c 4, are pan-cyclic. In this paper we show that all diregular c-partite tournaments, with c 5, are in fact vertex-pancyclic.

Almost regular c-partite tournaments with c ≥ 8 contain an n-cycle through a given arc for 4 ≤ n ≤ c

Almost all Regular Graphs are Hamiltonian

In a previous paper the authors showed that almost all labelled cubic graphs are hamiltonian. In the present paper, this result is used to show that almost all r-regular graphs are hamiltonian for any fixed r ≥ 3, by an analysis of the distribution of 1-factors in random regular graphs. Moreover, almost all such graphs are r-edge-colourable if they have an even number of vertices. Similarly, al...

Packing edge-disjoint triangles in regular and almost regular tournaments

For a tournament T , let ν3(T ) denote the maximum number of pairwise edge-disjoint triangles (directed cycles of length 3) in T . Let ν3(n) denote the minimum of ν3(T ) ranging over all regular tournaments with n vertices (n odd). We conjecture that ν3(n) = (1 + o(1))n /9 and prove that n 11.43 (1− o(1)) ≤ ν3(n) ≤ n 9 (1 + o(1)) improving upon the best known upper bound of n −1 8 and lower bou...