H-Cycles in H-Colored Multigraphs

Let H be a graph possibly with loops and G a multigraph without loops. G is said to be H -colored if there exists a function c : E ( G ) → V ( H ). A cycle ( v 0 , e 0 , v 1 , e 1 , … , e k - 1 , v k = v 0 ) in G , where e i = v i v i + 1 for every i in {0, … , k - 1 }, is an H -cycle if and only if ( c ( e 0 ), a 0 , c ( e 1 ), … , c ( e k - 2 ), a k - 2 , c ( e k - 1 ), a k - 1 , c ( e 0 )) is a walk in H , with a i = c ( e i ) c ( e i + 1 ) for every i in {0, … , k - 1 } (indices modulo k ). If H is a complete graph without loops, an H -walk is called properly colored walk. The problem of check whether an edge-colored graph G contains a properly colored cycle was studied first by Grossman and Häggkvist. Subsequently Yeo gave a sufficient condition which guarantee the existence of a properly colored cycle. In this paper we will extend Yeo’s result for the case where stronger requirements are enforced for a properly colored cycle to be eligible, based on the adjacencies of a graph whose vertices are in bijection with the colors. The main result establishes that if H is a graph without loops and G is an H -colored multigraph such that (1) H and G have no isolated vertices, (2) G has no H -cycles and (3) for every x in V ( G ), G x is a complete k x -partite graph for some k x in N . Then there exists a vertex z in V ( G ) such that every connected component D of G - z satisfies that { e ∈ E ( G ) : e = z u for some u in V ( D )} is an independent set in G z (where for w in V ( G ), G w is an associated graph to the vertex w , respect to the H -coloring of G ).