Interval non-edge-colorable bipartite graphs and multigraphs

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 Erd\H{o}s constructed a bipartite graph with 27 vertices and maximum degree 13 which has no interval coloring. Erd\H{o}s'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.Comment: 18 pages, 7 figure

Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees

An edge coloring of a graph G is called interval edge coloring if for each v ? V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by ?\u27_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (?,?)-biregular if each vertex in one part has degree ? and each vertex in the other part has degree ?. A graph G is called (?*,?*)-bipartite if G is a subgraph of an (?,?)-biregular graph and the maximum degree in one part is ? and the maximum degree in the other part is ?. In the paper we study the problem of interval edge colorings of (k*,2*)-bipartite graphs, for k ? {3,4,5}, and of (5*,3*)-bipartite graphs. We prove that every (5*,2*)-bipartite graph admits an interval edge coloring using at most 6 colors, which can be found in O(n^{3/2}) time, and we prove that an interval edge 5-coloring of a (5*,2*)-bipartite graph can be found in O(n^{3/2}) time, if it exists. We show that every (4^*,2^*)-bipartite graph admits an interval edge 4-coloring, which can be found in O(n) time. The two following problems of interval edge coloring are known to be NP-complete: 6-coloring of (6,3)-biregular graphs (Asratian and Casselgren (2006)) and 5-coloring of (5*,5*)-bipartite graphs (Giaro (1997)). In the paper we prove NP-completeness of 5-coloring of (5*,3*)-bipartite graphs

Some results on the palette index of graphs

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices