Generalizing Vertex Pancyclic and $k$-ordered Graphs

Abstract

Let $k\le m\le n$ be fixed positive integers. A graph of order $n$ is $(k,m)$-pancyclic if for any set of $k$ vertices and any integer $r$ with $m\le r\le n$, there is a cycle of length $r$ containing the $k$ vertices. If the additional property that the $k$ vertices must appear on the cycle in a specified order is required, then the graph is said to be $(k,m)$-pancyclic ordered. Faudree et al. (Graphs Comb 20:291–309, 2004) gave the condition of the minimum sum of degree of two nonadjacent vertices that implies a graph to be $(k,m)$-pancyclic or $(k,m)$-pancyclic ordered. In this paper, we introduce a stronger related property, $(k,m)$-vertex-pancyclic ordered graphs, which requires for any specified vertex $v$ and any ordered set $S$ of $k$ vertices there is a cycle of length $r$ containing $v$ and $S$ and encountering the vertices of $S$ in the specified order for each $m\le r\le n$. The condition of the minimum sum of degree of two nonadjacent vertices that implies a graph is $(k,m+2)$-vertex-pancyclic ordered are presented. Examples introduced by Faudree et al. also show that these constraints are best possible.