Generalizing Vertex Pancyclic and $$k$$ k -ordered Graphs

Abstract

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