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50101 Pure Mathematics
4Graph theory
2(k,g)-cage
2Number theory
2Numerical methods
2Theorem proving
2Vertex connectivity
10802 Computation Theory and Mathematics
1Algorithms
1Cages
1Cube
1Cubical polytope
1Cutset
1Edge connectivity
1Edge-connectivity
1Edge-superconnected
1Function evaluation
1Girth pair
1Hypercube

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A lower bound on the order of regular graphs with given girth pair

- Balbuena, Camino, Jiang, T., Lin, Yuqing, Marcote, Xavier, Miller, Mirka

**Authors:**Balbuena, Camino , Jiang, T. , Lin, Yuqing , Marcote, Xavier , Miller, Mirka**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 55, no. 2 (2007), p. 153-163**Full Text:**false**Reviewed:****Description:**The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 (1983), 209-218]. A (**Description:**C1**Description:**2003004727

All (k;g)-cages are k-edge-connected

- Lin, Yuqing, Miller, Mirka, Rodger, Chris

**Authors:**Lin, Yuqing , Miller, Mirka , Rodger, Chris**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 48, no. 3 (2005), p. 219-227**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)-cages are k-edge-connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)-cages are k-edge-connected if g is odd. Combining our results, we conclude that the (k;g)-cages are k-edge-connected. © 2005 wiley Periodicals, Inc.**Description:**C1

Improved lower bound for the vertex connectivity of (delta;g)-cages

- Lin, Yuqing, Miller, Mirka, Balbuena, Camino

**Authors:**Lin, Yuqing , Miller, Mirka , Balbuena, Camino**Date:**2005**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 299, no. 1-3 (Aug 2005), p. 162-171**Full Text:**false**Reviewed:****Description:**A (delta, g)-cage is a delta-regular graph with girth g and with the least possible number of vertices. We prove that all (delta, g)-cages are r-connected with r >= root(delta + 1) for g >= 7 odd. This result supports the conjecture of Fu, Huang and Rodger that all (delta; g)-cages are delta-connected. (c) 2005 Elsevier B.V. All rights reserved.**Description:**C1**Description:**2003001397

On the connectivity of (k, g)-cages of even girth

- Lin, Yuqing, Balbuena, Camino, Marcote, Xavier, Miller, Mirka

**Authors:**Lin, Yuqing , Balbuena, Camino , Marcote, Xavier , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 15 (2008), p. 3249-3256**Full Text:**false**Reviewed:****Description:**A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

All (k;g)-cages are edge-superconnected

- Lin, Yuqing, Miller, Mirka, Balbuena, Camino, Marcote, Xavier

**Authors:**Lin, Yuqing , Miller, Mirka , Balbuena, Camino , Marcote, Xavier**Date:**2006**Type:**Text , Journal article**Relation:**Networks Vol. 47, no. 2 (2006), p. 102-110**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is k-regular graph with girth g and with the least possible number of vertices. In this article we prove that (k;g)-cages are edge-superconnected if g is even. Earlier, Marcote and Balbuena proved that (k;g)-cages are edge-superconnected if g is odd [Networks 43 (2004), 54-59]. Combining our results, we conclude that all (k;g)-cages are edge-superconnected. © 2005 Wiley Periodicals, Inc.**Description:**C1**Description:**2003001830

Connectivity of cubical polytopes

- Bui, Hoa, Pineda-Villavicencio, Guillermo, Ugon, Julien

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2019**Type:**Text , Journal article , Article**Relation:**Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2019**Type:**Text , Journal article , Article**Relation:**Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.

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