Hajós' graph-coloring conjecture: Variations and counterexamples

Catlin, Paul A.

doi:10.1016/0095-8956(79)90062-5

Cited by 135 publications

(107 citation statements)

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“…Unfortunately, we still do not know if Hajós' conjecture is true in this case. However, for k ≥ 7, the conjecture was disproved by Catlin [Cat79], and shortly after Erdős and Fajtlowicz [ErF81] discovered that the conjecture combined with Turán's theorem [Tu41] would imply that every graph G with at least constant times k 3 vertices has k vertices that induce either a complete subgraph or an empty subgraph in G. (See also [Th05].) However, in his classic note [Er47] written 30 years earlier, Erdős used the "probabilistic method" to prove the existence of graphs with 2 k/2 vertices that do not have this property.…”

Section: Relaxations Of Planaritymentioning

confidence: 99%

The Beginnings of Geometric Graph Theory

Pach

2013

Bolyai Society Mathematical Studies

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Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straight-line edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal paper of Paul Erdős from 1946, we give a biased survey of Turán-type questions in the theory of geometric and topological graphs. What is the maximum number of edges that a geometric or topological graph of n vertices can have if it contains no forbidden subconfiguration of a certain type? We put special emphasis on open problems raised by Erdős or directly motivated by his work.

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Section: Relaxations Of Planaritymentioning

confidence: 99%

The Beginnings of Geometric Graph Theory

Pach

2013

Bolyai Society Mathematical Studies

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“…Chromatic Number: By [12,19], χ(C 2k+1 [r]) = 2r + r k . Thus, for a given r, for all but a finite number of small k, χ(C 2k+1 [r]) = 2r + 1.…”

Section: Cyclesmentioning

confidence: 99%

“…Plummer, Stiebitz, and Toft use the terminology r-inflation of C n [37], which is what we use in this article. In [12] Catlin gave a formula for the chromatic number of the r-inflation of C n . In [11], Boutin, Gethner, and Sulanke showed that for n ≥ 4 the 3-inflation of C n has thickness two.…”

Section: Introductionmentioning

confidence: 99%

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

2010

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The r-inflation of a graph G is the lexicographic product G with K r . A graph is said to have thickness t if its edges can be partitioned into t sets, each of which induces a planar graph, and t is smallest possible. In the setting of the r-inflation of planar graphs, we investigate the generalization of Ringel's famous Earth-Moon problem: What is the largest chromatic number of any thickness-t graph? In particular, we study classes of planar graphs for which we can determine both the thickness and chromatic number of their 2-inflations, and provide bounds on these parameters for their r-inflations. Moreover, in the same setting, we investigate arboricity and fractional chromatic number as well. We end with a list of open questions.

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“…At some point in the 40's, Hajós [17] conjectured that the relation of containment was the topological order. This conjecture is true for t ≤ 4 [10], but false for t ≥ 7 [4]. It remains open for t ∈ {5, 6}.…”

Section: Introductionmentioning

confidence: 98%

Complete graph immersions in dense graphs

Vergara

2017

Discrete Mathematics

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In this article we consider the relationship between vertex coloring and the immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston in 2003, which says that the complete graph with t vertices can be immersed in any t-chromatic graph, is studied.First, we present a general result about immersions and prove that the conjecture holds for graphs whose complement does not contain any induced cycle of length four and also for graphs having the property that every set of five vertices induces a subgraph with at least six edges.Then, we study the class of all graphs with independence number less than three, which are graphs of interest for Hadwiger's Conjecture. We study such graphs for the immersion-analog. If Abu-Khzam and Langston's conjecture is true for this class of graphs, then an easy argument shows that every graph of independence number less than 3 contains K ⌈ n 2 ⌉ as an immersion. We show that the converse is also true. That is, if every graph with independence number less than 3 contains an immersion of K ⌈ n 2 ⌉ , then Abu-Khzam and Langston's conjecture is true for this class of graphs. Furthermore, we show that every graph of independence number less than 3 has an immersion of K ⌈ n 3 ⌉ .

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Hajós' graph-coloring conjecture: Variations and counterexamples

Cited by 135 publications

References 4 publications

The Beginnings of Geometric Graph Theory

The Beginnings of Geometric Graph Theory

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

Complete graph immersions in dense graphs

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