On inclusion chromatic index of a graph

Abstract

Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident to any its neighbour. We provide an infinite family of examples of graphs $G$ with $\chi'_\subset(G)\geq (1+\frac{1}{\delta-1})\Delta$, where $\Delta$ is the maximum degree of $G$, and we conjecture that $\chi'_\subset(G)\leq \lceil(1+\frac{1}{\delta-1})\Delta\rceil$ for every connected graph with $\delta\geq 2$ which is not isomorphic to $C_5$. The equality here is attained e.g. for the family of complete bipartite graphs. Using a probabilistic argument we support this conjecture by proving that for any fixed $\delta\ge2$, $\chi'_\subset(G) \le (1+\frac{4}{\delta})\Delta (1+o(1))$ (for $\Delta\to\infty$), what implies that $\chi'_\subset(G) \le (1+\frac{4}{\delta-1})\Delta$ for $\Delta$ large enough.