Three topics in online list coloring

Carraher, James M.; Loeb, Sarah; Mahoney, Thomas; Puleo, Gregory J.; Tsai, Mu-Tsun; West, Douglas B.

doi:10.4310/joc.2014.v5.n1.a5

Cited by 10 publications

(12 citation statements)

References 16 publications

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“…The result on blocks [2] implies that it suffices to consider 2-connected graphs. Towards this conjecture, we show that a minimal counterexample G to Conjecture 1.1 must avoid several reducible structures.…”

Section: Conjecture 11 Every Outerplanar Graph Is Sp-greedymentioning

confidence: 94%

“…In Lemma 3.5 and Theorem 4.1, we apply Lemma 2.2 with a larger deleted set. It has been noted before [2] that we may assume Painter always responds by coloring a maximal independent subset of the marked set. We use this fact when dealing with larger deleted sets by only considering maximal independent subsets of each marked set.…”

Section: Corollaries 23 and 24 Imply That Ifmentioning

confidence: 99%

“…This notation was first used in[2]; it evokes the additivity of the vertex sets, it avoids conflicting with the proper use of "+" for disjoint union, and it is consistent with the "Czech notation" introduced by Nešetřil in which the notation displays the result of the operation on K 2 and K 2 .…”

mentioning

confidence: 96%

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Families of Online Sum-Choice-Greedy Graphs

Mahoney

Tomlinson

Wise

2014

Graphs and Combinatorics

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In online list coloring (introduced by Zhu and by Schauz in 2009), at each step a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of the marked vertices to receive that color. A graph G is said to be f -paintable for a function f : V (G) → N if there is an algorithm to produce a successful coloring whenever each vertex v is allowed to be marked at most f (v) times. In 2002 Isaak introduced sum list coloring and the resulting parameter called sum-choosability. The analogous notion of online sum-choosability, or sum-paintability, is the minimum of f (v) over all functions f such that G is f -paintable; we denote this value by χ sp (G). Always χ sp (G) ≤ |V (G)| + |E(G)|, and we say that G is sp-greedy when equality holds. We conjecture that all outerplanar graphs are sp-greedy. We prove this for every outerplanar graph whose weak dual is a path and give further restrictions on the structure of a minimal counterexample. We also prove that wheels are sp-greedy.

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Section: Conjecture 11 Every Outerplanar Graph Is Sp-greedymentioning

confidence: 94%

Section: Corollaries 23 and 24 Imply That Ifmentioning

confidence: 99%

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Families of Online Sum-Choice-Greedy Graphs

Mahoney

Tomlinson

Wise

2014

Graphs and Combinatorics

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“…Adding a closed ear is adding a cycle with one vertex in G. To study χ sp under addition of ears, we need an observation about degeneracy that is used in [2] and in earlier papers on paintability such as [7,8,13]. It states that vertices with "excess" tokens are irrelevant in determining whether a graph is f -paintable.…”

Section: Corollary 25 a Graph G Is Sp-greedy If And Only If Each Ofmentioning

confidence: 99%

“…Strengthening this result, we prove χ sp (K 2,r ) = χ sc (K 2,r ) by showing that Painter can win when lm > r, each vertex in the large part has 2 tokens, and the two vertices in the small part have l and m tokens. This follows from a result in [2] that determines who wins in a class of Lister/Painter games of K k,r .…”

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confidence: 99%

Sum-Paintability of Generalized Theta-Graphs

Carraher

Mahoney

Puleo

et al. 2014

Graphs and Combinatorics

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In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset of this set to receive that color. For a graph G and a function f : V (G) → N, the graph is f -paintable if there is an algorithm to produce a proper coloring when each vertex v is allowed to be presented at most f (v) times.The sum-paintability. . , H k are the blocks of G. Also, adding an ear of length ℓ to G adds 2ℓ − 1 to the sum-paintability, when ℓ ≥ 3.Strengthening a result of Berliner et al., we prove χ sp (K 2,r ) = 2r+min{l+m : lm > r}. The generalized theta-graph Θ ℓ 1 ,...,ℓ k consists of two vertices joined by internally disjoint paths of lengths ℓ 1 , . . . , ℓ k . A book is a graph of the form Θ 1,2,...,2 , denoted B r when there are r internally disjoint paths of length 2. We prove χ sp (B r ) = 2r + min l,m∈N {l + m : m(l − m) + m 2 > r}. We use these results to determine the sumpaintability for all generalized theta-graphs.

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Brooks' Theorem and Beyond

Cranston

Rabern²

2014

Journal of Graph Theory

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Abstract. We collect some of our favorite proofs of Brooks' Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon-Tarsi orientations, since analogues of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin-Kostochka Conjecture and Reed's Conjecture.Brooks' Theorem is among the most fundamental results in graph coloring. In short, it characterizes the (very few) connected graphs for which an obvious upper bound on the chromatic number holds with equality. It has been proved and reproved using a wide range of techniques, and the different proofs generalize and extend in many directions. In this paper we share some of our favorite proofs. In addition to surveying Brooks' Theorem, we aim to illustrate many of the standard techniques in vertex coloring 1 ; furthermore, we prove versions of Brooks' Theorem for standard strengthenings of vertex coloring, including list coloring, online list coloring, and Alon-Tarsi orientations. We present the proofs roughly in order of increasing complexity, but each section is self-contained and the proofs can be read in any order. Before we state the theorem, we need a little background.A proper coloring assigns colors, denoted by positive integers, to the vertices of a graph so that endpoints of each edge get different colors. A graph G is k-colorable if it has a proper coloring with at most k colors, and its chromatic number χ(G) is the minimum value k such that G is k-colorable. If a graph G has maximum degree ∆, then χ(G) ≤ ∆ + 1, since we can repeatedly color an uncolored vertex with the smallest color not already used on its neighbors. Since the proof of this upper bound is so easy, it is natural to ask whether we can strengthen it. The answer is yes, nearly always.A clique is a subset of vertices that are pairwise adjacent. If G contains a clique K k on k vertices, then χ(G) ≥ k, since all clique vertices need distinct colors. Similarly, if G contains an odd length cycle, then χ(G) ≥ 3, even when ∆ = 2. In 1941, Brooks [10] proved that these are the only two cases in which we cannot strengthen our trivial upper bound on χ(G).

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Three topics in online list coloring

Cited by 10 publications

References 16 publications

Families of Online Sum-Choice-Greedy Graphs

Families of Online Sum-Choice-Greedy Graphs

Sum-Paintability of Generalized Theta-Graphs

Brooks' Theorem and Beyond

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