Planar graphs without 5-cycles and intersecting triangles are $(1,1,0)$-colorable

TL;DR

This paper proves that planar graphs without 5-cycles and intersecting triangles are $(1,1,0)$-colorable, advancing understanding of graph colorability under specific structural constraints.

Contribution

It establishes that such graphs are $(1,1,0)$-colorable, providing a partial confirmation of Borodin and Raspaud's conjecture.

Findings

Graphs are $(1,1,0)$-colorable under given conditions

Supports Borodin and Raspaud's conjecture partially

Advances structural graph coloring theory

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 $5$-cycles and intersecting triangles is $(0,0,0)$-colorable. We prove in this paper that such graphs are $(1,1,0)$-colorable.