On Edge-Colored Graphs Covered by Properly Colored Cycles

Abstract. We characterize edge-colored graphs in which every edge belongs to some properly colored cycle. We obtain our result by applying a characterization of 1-extendable graphs. Key words. Edge-colored graphs, properly colored cycles, 1-extendable graphs, cycle covers. 1. Introduction, notation and statement of the main result All graphs considered are finite and have no loops; multiple edges, however, are allowed. For a graph G and X, Y ⊆ V (G) we denote the set of edges of G joining a vertex in X and a vertex in Y by E(X, Y); for v ∈ V (G) we put E(v): = E({v}, V (G)). For E ′ ⊆ E(G) we denote by V (E ′ ) the set of all v ∈ V (G) with E(v) ∩ E ′ � = ∅. Further graph theoreti