Vertex alternating-pancyclism in 2-edge-colored generalized sums of graphs

摘要

An alternating cycle in a 2-edge-colored graph is a cycle such that any two consecutive edges have different colors. Let G(1), ..., G(k) be a collection of pairwise vertex disjoint 2-edge-colored graphs. The colored generalized sum of G(1), ..., G(k) denoted by circle plus(k)(i=1)G(i), is the set of all 2-edge-colored graphs G such that: (i) V(G) = boolean OR(k)(i=1) V(G(i)), G < V(G(i))> congruent to G(i) for i = 1, ..., k as edge-colored graphs where G < V(G(i))> has the same coloring as Gi and (iii) between each pair of vertices in different summands of G there is exactly one edge, with an arbitrary but fixed color. A graph G in circle plus(k)(i=1)G(i) will be called a colored generalized sum (c.g.s.). A 2-edge-colored graph G of order 2n is vertex alternating-pancyclic iff, for each vertex v E V(G) and each k is an element of {2, ..., n}, G contains an alternating cycle of length 2k passing through v.