2013/01/16 by Petros A. Petrosyan, Petrosyan, Petros A., Hrant H. Khachatrian +1
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.DM #math.CO
paper · pdf · doi:10.48550/arxiv.1301.3811
18 pages, 7 figures
arxiv created 2013/01/18 · arxiv updated 2013/01/21
An edge-coloring of a graph G with colors 1,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991 Erdős constructed a bipartite graph with 27 vertices and maximum degree 13 which has no interval coloring. Erdős's counterexample is the smallest (in a sense of maximum degree) known bipartite graph which is not interval colorable. On the other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this paper we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs which have no interval coloring, contain 20,19,21 vertices and have maximum degree 11,12,13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.