Bonds Intersecting Cycles In A Graph

McGuinness, Sean

doi:10.1007/s00493-005-0026-6

Cited by 4 publications

(4 citation statements)

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“…In [2] we prove that: Theorem 2.1. For a k-connected graph G (k ≥ 2) having circumference c ≥ 2k, for any pair of cycles C 1 and C 2 which intersect in at most one vertex, it holds that |V (C 1 )| + |V (C 2 )|≤ 2(c − k + 1).…”

Section: Maximum Cycles and Bonds In A Graphmentioning

confidence: 80%

“…Theorem 2.1 implies (see [2]): Theorem 2.2. For any k-connected graph G (k ≥ 2) having circumference c ≥ 2k, there is a bond B which intersects every cycle of length c − k + 2 or greater.…”

Section: Maximum Cycles and Bonds In A Graphmentioning

confidence: 89%

“…To begin with, we briefly define some terminology to be used in this section, and refer the reader to [6] for further reference. A matroid is binary if it is representable over GF (2), and it is regular if it is representable over every field. Let M be matroid and let k be a positive…”

Section: Extensions To Matroidsmentioning

confidence: 99%

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Circuits Through Cocircuits In A Graph With Extensions To Matroids

McGuinness

2005

Combinatorica

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We show that for any k-connected graph having cocircumference c * , there is a cycle which intersects every cocycle of size c * − k + 2 or greater. We use this to show that in a 2connected graph, there is a family of at most c * cycles for which each edge of the graph belongs to at least two cycles in the family. This settles a question raised by Oxley.A certain result known for cycles and cocycles in graphs is extended to matroids. It is shown that for a k-connected regular matroid having circumference c ≥ 2k if C1 and C2 are disjoint circuits satisfying r(C1) + r(C2) = r(C1 ∪ C2), then |C1| + |C2| ≤ 2(c − k + 1).

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Section: Maximum Cycles and Bonds In A Graphmentioning

confidence: 80%

Section: Maximum Cycles and Bonds In A Graphmentioning

confidence: 89%

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Circuits Through Cocircuits In A Graph With Extensions To Matroids

McGuinness

2005

Combinatorica

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“…Theorem 1.2 ( [15]). Let G be a k-connected graph with longest cycle of length c, where c ≥ 2k and k ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

Bonds intersecting long paths in $k$-connected graphs

Wei¹,

H²,

Zhao³

2022

Preprint

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A well-known question of Gallai (1966) [9] asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs [1,6,13,17], the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex v if and only if it meets the vertex-bond with respect to v. Therefore, a very natural approach to Gallai's question is to study whether there is a bond meeting all longest paths. Let p denote the length of a longest path of connected graphs. We show that for any 2-connected graph, there is a bond meeting all paths of length at least p − 1. We then prove that for any 3-connected graph, there is a bond meeting all paths of length at least p − 2. For a k-connected graph (k ≥ 3), we show that there is a bond meeting all paths of length at least p − t + 1, where t = k−2 2 if p is even and t = k−2 2 if p is odd. Our results provide analogs of the corresponding results of P. Wu [15] and S. McGuinness [22, Bonds intersecting cycles in a graph, Combinatorica 25 (4) (2005), 439-450] also.

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