A relaxation of the Bordeaux Conjecture

TL;DR

This paper relaxes a longstanding conjecture by proving that planar graphs without intersecting triangles and 5-cycles are (2,0,0)-colorable, advancing understanding of graph colorability under specific constraints.

Contribution

It introduces a new coloring result for planar graphs, showing they are (2,0,0)-colorable under certain cycle restrictions, thus partially confirming a relaxed version of the Bordeaux Conjecture.

Findings

Proves planar graphs without intersecting triangles and 5-cycles are (2,0,0)-colorable.

Provides a partial validation of a relaxed Bordeaux Conjecture.

Advances graph coloring theory for specific planar graph classes.

Abstract

A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $\varphi:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored $i$. Borodin and Raspaud conjecture that every planar graph without intersecting triangles and $5$-cycles is $3$-colorable. We prove in this paper that every planar graph without intersecting triangles and $5$-cycles is (2,0,0)-colorable.