On a special case of Hadwiger's conjecture

Plummer, Michael D.; Stiebitz, Michael; Toft, Bjarne

doi:10.7151/dmgt.1206

Cited by 46 publications

(39 citation statements)

References 12 publications

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“…However, the two conjectures are actually equivalent. Following ideas from [20] we show the next result. A weaker version of Conjecture 3 is shown, namely the following result.…”

Section: Conjecture 1 (Abu-khzam and Langstonmentioning

confidence: 88%

“…Finally, we focus on the study of a special class of graphs, the graphs G that have no independent set of size three, or equivalently, whose independence number α(G) is at most 2. This class of graphs has been extensively studied in an attempt to solve Hadwiger's conjecture (see [2,5,6,20]). It is for this reason that we are interested in Abu-Khzam and Langston's conjecture restricted to these graphs.…”

Section: Conjecture 1 (Abu-khzam and Langstonmentioning

confidence: 99%

“…The proof of Theorem 1.5 is strongly inspired from [20]. We will need to use some preliminary results.…”

Section: Equivalence Of Conjectures 2 Andmentioning

confidence: 99%

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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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“…However, the two conjectures are actually equivalent. Following ideas from [20] we show the next result. A weaker version of Conjecture 3 is shown, namely the following result.…”

Section: Conjecture 1 (Abu-khzam and Langstonmentioning

confidence: 88%

Section: Conjecture 1 (Abu-khzam and Langstonmentioning

confidence: 99%

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Complete graph immersions in dense graphs

Vergara

2017

Discrete Mathematics

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“…Thomassen describes these gaphs as the r-uniform replication of C n [42]. 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 .…”

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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“…Toft gave this case a thorough treatment in [30]. Despite many partial and tangential results on this case, the bound in Theorem 3 is still the best known for a = 2 (Note that for small values of a, the bound in Theorem 3 is better than the bound in Theorem 5).…”

Section: Theorem 4 For Any Graph G On N Vertices Withmentioning

confidence: 99%

Connected matchings in special families of graphs.

Caragianis¹

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On a special case of Hadwiger's conjecture

Abstract: Hadwiger's Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α(G) = 2. We present some results in this special case.

Cited by 46 publications

References 12 publications

Complete graph immersions in dense graphs

Complete graph immersions in dense graphs

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

Connected matchings in special families of graphs.

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