A relaxation of the Bordeaux Conjecture

Abstract

A $(c_1,c_2,\dots ,c_k)$-coloring of a graph $G$ is a mapping $\phi :V\left(G\right)\mapsto \{1,2,\dots ,k\}$ such that for every $i,1\le i\le k$, $G\left[V_i\right]$ has maximum degree at most $c_i$, where $G\left[V_i\right]$ denotes the subgraph induced by the vertices colored $i$. Borodin and Raspaud conjecture that every planar graph with neither $5$-cycles nor intersecting triangles is $3$-colorable. We prove in this paper that every planar graph with neither $5$-cycles nor intersecting triangles is (2, 0, 0)-colorable.