Cooperative colorings and independent systems of representatives
The electronic journal of combinatorics, Tome 22 (2015) no. 2
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.
@article{10_37236_2488,
author = {Ron Aharoni and Ron Holzman and David Howard and Philipp Spr\"ussel},
title = {Cooperative colorings and independent systems of representatives},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {2},
doi = {10.37236/2488},
zbl = {1325.05073},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2488/}
}
TY - JOUR AU - Ron Aharoni AU - Ron Holzman AU - David Howard AU - Philipp Sprüssel TI - Cooperative colorings and independent systems of representatives JO - The electronic journal of combinatorics PY - 2015 VL - 22 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.37236/2488/ DO - 10.37236/2488 ID - 10_37236_2488 ER -
%0 Journal Article %A Ron Aharoni %A Ron Holzman %A David Howard %A Philipp Sprüssel %T Cooperative colorings and independent systems of representatives %J The electronic journal of combinatorics %D 2015 %V 22 %N 2 %U http://geodesic.mathdoc.fr/articles/10.37236/2488/ %R 10.37236/2488 %F 10_37236_2488
Ron Aharoni; Ron Holzman; David Howard; Philipp Sprüssel. Cooperative colorings and independent systems of representatives. The electronic journal of combinatorics, Tome 22 (2015) no. 2. doi: 10.37236/2488
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