Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree Δ, then G is entirely 7-choosable if Δ≤4 and G is entirely (Δ+ 2)-choosable if Δ≥ 5; that is, if every vertex, edge and face of G is given a list of max{7,Δ+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K2,3-minor-free graph or a (K2 + (K1 U K2))-minor-free graph. As a special case this proves that the Entire Colouring Conjecture, that a plane graph is entirely (Δ + 4)-colourable, holds if G is a plane embedding of a K4-minor-free graph, a K2,3-minor-free graph or a (K2 + (K1 U K2))-minor-free graph

Extending List Colorings of Planar Graphs

In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ have been assigned colors from their respective lists. We give a new, simplified proof where there are a small number of precolored vertices on the same face. We also explore cases where $W=\{u,v\}$ and $G$ has a separating $C_3$ or $C_4$ between $u$ and $v$

Injective choosability of subcubic planar graphs with girth 6

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor have distinct colors. A graph G is injectively k-choosable if for any list assignment L, where |L(v)| ≥ k for all v ∈ V(G), G has an injective L-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens the result of Lužar, Škrekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases

Circular choosability

International audienceWe study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O(ch(G) + ln |V(G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f-choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of τ := sup {cch(G) : G is planar}, and we prove that 68, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ℓ ∈ {3, 4, 5, 6, 7}, a planar graph is 4-choosable if it is ℓ-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ ∈ {5, 6, 7} that a planar graph is (4, 2)-choosable if it does not contain chorded ℓ-cycles

List-colourings of near-outerplanar graphs

A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a graph is outerplanar if it is both K4-minor-free and K2,3-minor-free, then by a near-outerplanar graph we mean a graph that is either K4-minor-free or K2,3-minor-free. Chapter 1 gives an introduction to the area of graph colourings, and includes a review of several results and conjectures in this area. In particular, four important and interesting conjectures in graph theory are the List-Edge-Colouring Conjecture (LECC), the List-Total-Colouring Conjecture (LTCC), the Entire Colouring Conjecture (ECC), and the List-Square-Colouring Conjecture (LSCC), each of which will be discussed in Chapter 1. In Chapter 2 we include a proof of the LECC and LTCC for all near-outerplanar graphs. In Chapter 3 we will study the list-colouring of a near-outerplanar graph in which vertices and faces, edges and faces, or vertices, edges and face are to be coloured. The results for the case when all elements are to be coloured will prove the ECC for all near-outerplanar graphs. In Chapter 4 we will study the list-colouring of the square of a K4-minor-free graph, and in Chapter 5 we will study the list-colouring of the square of a K2,3-minor-free graph. In Chapter 5 we include a proof of the LSCC for all K2,3-minor-free graphs with maximum degree at least six

Online choosability of graphs

We study several problems in graph coloring. In list coloring, each vertex $v$ has a set $L(v)$ of available colors and must be assigned a color from this set so that adjacent vertices receive distinct colors; such a coloring is an $L$-coloring, and we then say that $G$ is $L$-colorable. Given a graph $G$ and a function $f:V(G)\to\N$, we say that $G$ is $f$-choosable if $G$ is $L$-colorable for any list assignment $L$ such that $|L(v)|\ge f(v)$ for all $v\in V(G)$. When $f(v)=k$ for all $v$ and $G$ is $f$-choosable, we say that $G$ is $k$-choosable. The least $k$ such that $G$ is $k$-choosable is the choice number, denoted $\ch(G)$. We focus on an online version of this problem, which is modeled by the Lister/Painter game. The game is played on a graph in which every vertex has a positive number of tokens. In each round, Lister marks a nonempty subset $M$ of uncolored vertices, removing one token at each marked vertex. Painter responds by selecting a subset $D$ of $M$ that forms an independent set in $G$. A color distinct from those used on previous rounds is given to all vertices in $D$. Lister wins by marking a vertex that has no tokens, and Painter wins by coloring all vertices in $G$. When Painter has a winning strategy, we say that $G$ is $f$-paintable. If $f(v)=k$ for all $v$ and $G$ is $f$-paintable, then we say that $G$ is $k$-paintable. The least $k$ such that $G$ is $k$-paintable is the paint number, denoted \pa(G). In Chapter 2, we develop useful tools for studying the Lister/Painter game. We study the paintability of graph joins and of complete bipartite graphs. In particular, \pa(K_{k,r})\le k if and only if $r<k^k$. In Chapter 3, we study the Lister/Painter game with the added restriction that the proper coloring produced by Painter must also satisfy some property $\mathcal{P}$. The main result of Chapter 3 provides a general method to give a winning strategy for Painter when a strategy for the list coloring problem is already known. One example of a property $\mathcal{P}$ is that of having an $r$-dynamic coloring, where a proper coloring is $r$-dynamic if each vertex $v$ has at least $\min\set{r,d(v)}$ distinct colors in its neighborhood. For any graph $G$ and any $r$, we give upper bounds on how many tokens are necessary for Painter to produce an $r$-dynamic coloring of $G$. The upper bounds are in terms of $r$ and the genus of a surface on which $G$ embeds. In Chapter 4, we study a version of the Lister/Painter game in which Painter must assign $m$ colors to each vertex so that adjacent vertices receive disjoint color sets. We characterize the graphs in which $2m$ tokens is sufficient to produce such a coloring. We strengthen Brooks' Theorem as well as Thomassen's result that planar graphs are 5-choosable. In Chapter 5, we study sum-paintability. The sum-paint number of a graph $G$, denoted \spa(G), is the least $\sum f(v)$ over all $f$ such that $G$ is $f$-paintable. We prove the easy upper bound: \spa(G)\le|V(G)|+|E(G)|. When \spa(G)=|V(G)|+|E(G)|, we say that $G$ is sp-greedy. We determine the sum-paintability of generalized theta-graphs. The generalized theta-graph $\Theta_{\ell_1,\dots,\ell_k}$ consists of two vertices joined by $k$ paths of lengths \VEC \ell1k. We conjecture that outerplanar graphs are sp-greedy and prove several partial results toward this conjecture. In Chapter 6, we study what happens when Painter is allowed to allocate tokens as Lister marks vertices. The slow-coloring game is played by Lister and Painter on a graph $G$. Lister marks a nonempty set of uncolored vertices and scores 1 point for each marked vertex. Painter colors all vertices in an independent subset of the marked vertices with a color distinct from those used previously in the game. The game ends when all vertices have been colored. The sum-color cost of a graph $G$, denoted \scc(G), is the maximum score Lister can guarantee in the slow-coloring game on $G$. We prove several general lower and upper bounds for \scc(G). In more detail, we study trees and prove sharp upper and lower bounds over all trees with $n$ vertices. We give a formula to determine \scc(G) exactly when $\alpha(G)\le2$. Separately, we prove that \scc(G)=\spa(G) if and only if $G$ is a disjoint union of cliques. Lastly, we give lower and upper bounds on \scc(K_{r,s})