3-coloring triangle-free planar graphs with a precolored 8-cycle

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.Comment: 20 pages, 5 figure

On the Strong Parity Chromatic Number

International audienceA vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs

Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio

Coloring problems in graph theory

In this thesis, we focus on variants of the coloring problem on graphs. A coloring of a graph $G$ is an assignment of colors to the vertices. A coloring is proper if no two adjacent vertices are assigned the same color. Colorings are a central part of graph theory and over time many variants of proper colorings have been introduced. The variants we study are packing colorings, improper colorings, and facial unique-maximum colorings. A packing coloring of a graph $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that the distance between any two vertices that receive color $i$ is greater than $i$. A $(d_1, \ldots, d_k)$-coloring of $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that the distance between any two vertices that receive color $i$ is greater than $d_i$. We study packing colorings of multi-layer hexagonal lattices, improving a result of Fiala, Klav\v{z}ar, and Lidick\\u27{y}, and find the packing chromatic number of the truncated square lattice. We also prove that subcubic planar graphs are $(1, 1, 2, 2, 2)$-colorable. A facial unique-maximum coloring of $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that no two adjacent vertices receive the same color and the maximum color on a face appears only once on that face. We disprove a conjecture of Fabrici and G\ {o}ring that plane graphs are facial unique-maximum $4$-colorable. Inspired by this result, we also provide sufficient conditions for the facial unique-maximum $4$-colorability of a plane graph. A $\{ 0, p \}$-coloring of $G$ is an assignment of colors $0$ and $p$ to the vertices of $G$ such that the vertices that receive color $0$ form an independent set and the vertices that receive color $p$ form a linear forest. We will explore $\{ 0, p \}$-colorings, an offshoot of improper colorings, and prove that subcubic planar $K_4$-free graphs are $\{ 0, p \}$-colorable. This result is a corollary of a theorem by Borodin, Kostochka, and Toft, a fact that we failed to realize before the completion of our proof