Matchings and Hadwiger's Conjecture

Kotlov, Andrei

doi:10.1016/s0012-365x(01)00087-5

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…We now know the structure of G up to vertex c-duplication-G is an inflation of the complement of the Petersen graph ((ii) holds). This proves (2 ). If (2 )(i) holds, then G\(M 1 ∪ N 1 ) and G\(M 2 ∪ N 2 ) are disconnected.…”

Section: Conclusion Conjectures and Future Worksupporting

confidence: 52%

“…However, we have quite a bit of control on the counterexamples. Note that the proof of Theorem 3.2 mentions the good edge axiom and uses the special choices (I) and (II) for the partition only in step (2). By an easy argument from the proof of Theorem A.1, any counterexample must have exactly two components of medium edges, L and R, with at least two dipoles in each.…”

Section: Extensionsmentioning

confidence: 99%

“…The precise statement is Proof. We will use steps (1) and (3) of the proof of Theorem 3.2, which do not depend on constraints (I) and (II), and replace step (2) with what follows.…”

Section: Appendix Amentioning

confidence: 99%

“…Incidentally, the question of whether Hadwiger's conjecture holds for graphs with no stable set of size 3 and clique covering number 3 was posed by Andreȋ Kotlov in [2], although his paper was not known to us until after we completed this work. Section 3 describes the proof of our main result and some attempts at generalization.…”

Section: Introductionmentioning

confidence: 99%

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A special case of Hadwiger's conjecture

Blasiak

2007

Journal of Combinatorial Theory, Series B

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We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t − 1 vertices is not t − 1 colorable, so is conjectured to have a K t minor. There is a strengthening of Hadwiger's conjecture in this case, which states that there is a K t minor in which the preimage of each vertex of K t is a single vertex or an edge. We prove this strengthened version for graphs with an even number of vertices and fractional clique covering number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large n, a graph on n vertices with no stable set of size 3 has a K 1 9 n 4/5 minor using only vertices and single edges as preimages of vertices.

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Section: Conclusion Conjectures and Future Worksupporting

confidence: 52%

Section: Extensionsmentioning

confidence: 99%

“…The precise statement is Proof. We will use steps (1) and (3) of the proof of Theorem 3.2, which do not depend on constraints (I) and (II), and replace step (2) with what follows.…”

Section: Appendix Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A special case of Hadwiger's conjecture

Blasiak

2007

Journal of Combinatorial Theory, Series B

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“…The following theorem is well known in the literature regarding Hadwiger's conjecture (See Kotlov [17] for an alternate proof. )…”

Section: Proofmentioning

confidence: 98%

Hadwiger’s conjecture for proper circular arc graphs

Belkale

Chandran

2009

European Journal of Combinatorics

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Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc representation where no arc is completely contained in any other arc. Hadwiger's conjecture states that if a graph G has chromatic number k, then a complete graph on k vertices is a minor of G. We prove Hadwiger's conjecture for proper circular arc graphs.Definition 4 Complete graph of order n is a graph on n vertices where all the vertices are pairwise adjacent and is denoted by K n .See Diestel [2] for more information on minors.In 1943, Hadwiger made the famous conjecture linking the chromatic number of a graph with its clique minor:Conjecture 5 (Hadwiger [9]) If a graph G has chromatic number χ(G) = r, then K r is a minor of G.

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Schuldrecht

Recht Für Ingenieure

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Matchings and Hadwiger's Conjecture

Cited by 5 publications

References 7 publications

A special case of Hadwiger's conjecture

A special case of Hadwiger's conjecture

Hadwiger’s conjecture for proper circular arc graphs

Schuldrecht

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