Detecting minors in matroids through triangles

Albar, Boris; Alfonsín, Jorge Luis Ramírez

doi:10.1016/j.ejc.2015.10.010

Cited by 3 publications

(4 citation statements)

References 14 publications

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“…As opposed to graphs, the case where every element of the matroid belongs to three triangles is already intricate. This case has been settled in [2].…”

Section: Conjecture 82mentioning

confidence: 94%

“…By the second item of Proposition 6.3, ( ) ≥ + 1. By Theorems 1.5 and 1.7 (2), this graph has an edge that belongs to at most − 3 triangles. This contradicts the last item of Proposition 6.3.…”

Section: Conjecture 62 If Is a Double-critical -Chromatic Graph Thmentioning

confidence: 95%

“…Let us now give an alternative proof of the case = 8 that uses Theorem 1.7(1) instead of Theorem 1.7 (2). This approach might be useful to prove the next case of Conjecture 6.2.…”

Section: Conjecture 62 If Is a Double-critical -Chromatic Graph Thmentioning

confidence: 99%

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On triangles in ‐minor free graphs

Albar

2017

Journal of Graph Theory

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We study graphs where each edge that is incident to a vertex of small degree (of degree at most 7 and 9, respectively) belongs to many triangles (at least 4 and 5, respectively) and show that these graphs contain a complete graph (K6 and K7, respectively) as a minor. The second case settles a problem of Nevo. Moreover, if each edge of a graph belongs to six triangles, then the graph contains a K8‐minor or contains K2, 2, 2, 2, 2 as an induced subgraph. We then show applications of these structural properties to stress freeness and coloring of graphs. In particular, motivated by Hadwiger's conjecture, we prove that every K7‐minor free graph is 8‐colorable and every K8‐minor free graph is 10‐colorable.

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“…As opposed to graphs, the case where every element of the matroid belongs to three triangles is already intricate. This case has been settled in [2].…”

Section: Conjecture 82mentioning

confidence: 94%

Section: Conjecture 62 If Is a Double-critical -Chromatic Graph Thmentioning

confidence: 95%

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On triangles in ‐minor free graphs

Albar

2017

Journal of Graph Theory

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“…The main result of [15] implies that M is ternary. Theorem 2 of [1] implies that M P 7 or M has a minor isomorphic to a 3-wheel. For convenience, in this subsection we call a 3-connected matroid by strictly triangular matroid if each of its elements belongs to at least 2 triangles and every pair of triangles intersects in at most one element.…”

Section: Proof Of Theorem 15: Diamantic and Totally Triangular Matroidsmentioning

confidence: 99%

On the Structure of Triangle-Free 3-Connected Matroids

Santos¹

2021

Preprint

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Let $\mathcal{N}$ be an arbitrary class of matroids, closed under isomorphism. For $k$ a positive integer, we say that $M \in \mathcal{N}$ is \emph{$k$-minor-irreducible} if $M$ has no minor $N \in \mathcal{N}$ such that $1\leq\left|E\left(M\right)\right|-\left|E\left(N\right)\right|\leq k $. Tutte's Wheels and Whirls Theorem establish that, up to isomorphism, there are only two families of 1-minor-irreducible matroids in the class of 3-connected matroids. More recently, Lemos classified the 3-minor-irreducibles with at least 14 elements in the class of triangle-free 3-connected matroids. Here we prove a local characterization for the 2-minor-irreducible matroids with at least 11 elements in the class of triangle-free 3-connected matroids. This local characterization is used to establish two new families of 2-minor-irreducible matroids in this class.

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On the structure of triangle-free 3-connected matroids

Santos

2022

Advances in Applied Mathematics

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Detecting minors in matroids through triangles

Abstract: Abstract. In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of triangles, for a matroid to have either U 2,4 or F 7 or M (K 5 ) as a minor.

Cited by 3 publications

References 14 publications

On triangles in ‐minor free graphs

On triangles in ‐minor free graphs

On the Structure of Triangle-Free 3-Connected Matroids

On the structure of triangle-free 3-connected matroids

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