A relaxation of Steinberg's Conjecture

TL;DR

The paper proves that planar graphs without 4- and 5-cycles can be colored with specific degree constraints, relaxing Steinberg's Conjecture which posited proper 3-colorability for such graphs.

Contribution

It demonstrates new degree-bounded colorings for planar graphs excluding 4- and 5-cycles, extending the understanding of graph colorability under cycle restrictions.

Findings

Planar graphs without 4- and 5-cycles are (1,1,0)-colorable.

Such graphs are also (3,0,0)-colorable.

This relaxes the Steinberg Conjecture on proper 3-colorability.

Abstract

A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar graph without 4- and 5-cycles is $(1, 1, 0)$-colorable and $(3,0,0)$-colorable. This is a relaxation of the Steinberg Conjecture that every planar graph without 4- and 5-cycles are properly 3-colorable (i.e., $(0,0,0)$-colorable).