Defective correspondence coloring of planar graphs

TL;DR

This paper explores defective correspondence coloring of planar graphs, demonstrating limitations of 3-defective 3-correspondability and establishing that outerplanar graphs are 3-defective 2-correspondence colorable, with optimal defect bounds.

Contribution

It extends known results from list coloring to correspondence coloring, showing new limitations and capabilities in defective coloring of planar and outerplanar graphs.

Findings

Existence of a planar graph not 3-defective 3-correspondable.

Construction of a planar graph that is 1-defective 3-correspondable but not 4-correspondable.

All outerplanar graphs are 3-defective 2-correspondence colorable, with this bound being optimal.

Abstract

Defective coloring (also known as relaxed or improper coloring) is a generalization of proper coloring defined as follows: for $d \in \mathbb{N}$, a coloring of a graph is $d$-defective if every vertex is colored the same as at most $d$ of its neighbors. We investigate defective coloring of planar graphs in the context of correspondence coloring, a generalization of list coloring introduced by Dvo\v{r}\'ak and Postle. First we show there exists a planar graph that is not $3$-defective $3$-correspondable, strengthening a recent result of Cho, Choi, Kim, Park, Shan, and Zhu. Then we construct a planar graph that is $1$-defective $3$-correspondable but not $4$-correspondable, thereby extending a recent result of Ma, Xu, and Zhu from list coloring to correspondence coloring. Finally we show all outerplanar graphs are $3$-defective $2$-correspondence colorable, with 3 defects being best possible.