Circular Backbone Colorings: on matching and tree backbones of planar graphs

TL;DR

This paper investigates bounds on circular backbone colorings of planar graphs with specific subgraph restrictions, improving known upper bounds and addressing conjectures related to graph coloring.

Contribution

It provides new upper bounds for circular backbone chromatic numbers in planar graphs with certain subgraph constraints, advancing understanding of coloring problems and related conjectures.

Findings

If G is planar with no C4 and H is a linear spanning forest, then CBC_2(G,H) ≤ 7.

If G is a plane graph with no two 3-faces sharing an edge and H is a matching, then CBC_2(G,H) ≤ 6.

If G is planar with no C4 or C5 and H is a matching, then CBC_2(G,H) ≤ 5.

Abstract

Given a graph $G$, and a spanning subgraph $H$ of $G$, a circular $q$-backbone $k$-coloring of $(G,H)$ is a proper $k$-coloring $c$ of $G$ such that $q\le \lvert c(u)-c(v)\rvert \le k-q$, for every edge $uv\in E(H)$. The circular $q$-backbone chromatic number of $(G,H)$, denoted by $CBC_q(G,H)$, is the minimum integer $k$ for which there exists a circular $q$-backbone $k$-coloring of $(G,H)$. The Four Color Theorem implies that whenever $G$ is planar, we have $CBC_2(G,H)\le 8$. It is conjectured that this upper bound can be improved to 7 when $H$ is a tree, and to 6 when $H$ is a matching. In this work, we show that: 1) if $G$ is planar and has no $C_4$ as subgraph, and $H$ is a linear spanning forest of $G$, then $CBC_2(G,H)\leq 7$; 2) if $G$ is a plane graph having no two 3-faces sharing an edge, and $H$ is a matching of $G$, then $CBC_2(G,H)\leq 6$; and 3) if $G$ is planar and has no $C_4$ nor $C_5$ as subgraph, and $H$ is a mathing of $G$, then $CBC_2(G,H)\leq 5$. These results partially answer questions posed by Broersma, Fujisawa and Yoshimoto (2003), and by Broersma, Fomin and Golovach (2007). It also points towards a positive answer for the Steinberg's Conjecture.