Arc-pancyclicity of hypertournaments with irregularity at most two

Abstract

A k-hypertournament H consists of a pair (V(H),A(H)), where V(H) is a non-empty finite set of vertices and A(H) contains exactly one permutation of S as an arc for all k-subsets S of V(H). The irregularity of hypertournament H=(V(H),A(H)) is defined as maxv∈V(H)⁡{|deg+(v)−deg−(v)|}. A hypertournament with irregularity 0 is called as a ∑-regular or degree regular hypertournament which was introduced by Surmacs and is a generalization of regular tournament. A hypertournament with irregularity 1 is called as an almost ∑-regular or almost degree regular hypertournament which is a generalization of almost regular tournament.It is well known that every regular tournament is arc-pancyclic and every almost regular tournament with at least 8 vertices is arc-4-pancyclic. Surmacs (2017) [7] proved that every ∑-regular k-hypertournament on n-vertices, where 2≤k≤n−2, is arc-pancyclic. In this paper, we prove that every k-hypertournament on n vertices with irregularity at most two, where 3≤k≤n−2, is arc-pancyclic, which is a generalization of above results.