Cooperative Colorings and Independent Systems of Representatives

Abstract

We study a generalization of the notion of coloring of graphs, similar in spirit to that of list colorings: a cooperative coloring of a family of graphs $G_1,G_2, \ldots,G_k$ on the same vertex set $V$ is a choice of independent sets $A_i$ in $G_i$ ($1 \le i \le k)$ such that $\bigcup_{i=1}^kA_i=V$. This notion is linked (with translation in both directions) to the notion of ISRs, which are choice functions on given sets, whose range belongs to some simplicial complex. When the complex is that of the independent sets in a graph $G$, an ISR for a partition of the vertex set of a graph $G$ into sets $V_1,\ldots, V_n$ is a choice of a vertex $v_i \in V_i$ for each $i$ such that $\{v_1,\ldots,v_n\}$ is independent in $G$. Using topological tools, we study degree conditions for the existence of cooperative colorings and of ISRs. A sample result: Three cycles on the same vertex set have a cooperative coloring.