Proper path‐factors and interval edge‐coloring of (3,4)‐biregular bigraphs
Armen S. Asratian, Carl Johan Casselgren, Jennifer Vandenbussche, Douglas B. West · Journal of Graph Theory · 2009
Abstract An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)‐biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)‐biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3‐valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph. © 2009 Wiley Periodicals, Inc. J Graph Theory