scholarly journals On Snarks that are far from being 3-Edge Colorable

10.37236/3430 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jonas Hägglund
Keyword(s):  
Double Cover ◽  
Interesting Property ◽  
Perfect Matchings ◽  
Petersen Graph ◽  
Cubic Graph

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

scholarly journals The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

2012 ◽  
Vol 22 (3) ◽  
pp. 765-778
Author(s):  
Piotr Formanowicz ◽  
Krzysztof Tanaś
Keyword(s):  
Graph Theory ◽  
Computer Networks ◽  
Assignment Problem ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Algorithmic Approach ◽  
Edge Colorings ◽  
Mathematical Properties

Abstract It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan–Raspaud conjecture.

scholarly journals Double covers of graphs

1976 ◽  
Vol 14 (2) ◽  
pp. 233-248 ◽  
Author(s):  
Derek A. Waller
Keyword(s):  
Theoretical Approach ◽  
Upper Bound ◽  
Double Cover ◽  
Petersen Graph ◽  
Classification Theorem ◽  
Finite Graphs ◽  
Vertex Set ◽  
Double Covers

A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

scholarly journals On Essentially 4-Edge-Connected Cubic Bricks

10.37236/8594 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Nishad Kothari ◽  
Marcelo H. De Carvalho ◽  
Cláudio L. Lucchesi ◽  
Charles H. C. Little
Keyword(s):  
Petersen Graph ◽  
Triangular Prism ◽  
Cubic Graph

Lovász (1987) proved that every matching covered graph $G$ may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let $b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is quasi-$b$-invariant.) A brick $G$ is near-bipartite if it has a pair of edges $\{e,f\}$ so that $G-e-f$ is matching covered and bipartite; such a pair $\{e,f\}$ is a removable doubleton. (Each of $K_4$ and the triangular prism $\overline{C_6}$ has three removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture of Lovász which states that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge. A cubic graph is essentially $4$-edge-connected if it is $2$-edge-connected and if its only $3$-cuts are the trivial ones; it is well-known that each such graph is either a brick or a brace; we provide a graph-theoretical proof of this fact. We prove that if $G$ is any essentially $4$-edge-connected cubic brick then its edge-set may be partitioned into three (possibly empty) sets: (i) edges that participate in a removable doubleton, (ii) $b$-invariant edges, and (iii) quasi-$b$-invariant edges; our Main Theorem states that if $G$ has two adjacent quasi-$b$-invariant edges, say $e_1$ and $e_2$, then either $G$ is the Petersen graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of $G$ (distinct from $e_1$ and $e_2$) is $b$-invariant. As a corollary, we deduce that each essentially $4$-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges.

scholarly journals $b$-Invariant Edges in Essentially 4-Edge-Connected Near-Bipartite Cubic Bricks

10.37236/8945 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Fuliang Lu ◽  
Xing Feng ◽  
Yan Wang
Keyword(s):  
Lower Bound ◽  
Building Blocks ◽  
Petersen Graph ◽  
Cubic Graph ◽  
Bipartite Matching

A brick is a  non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is $b$-invariant if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is essentially 4-edge-connected if it does not contain nontrivial 3-cuts. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi  and Little proved that each essentially 4-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented.

scholarly journals Multiple Petersen Subdivisions in Permutation Graphs

10.37236/2239 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Tomáš Kaiser ◽  
Jean-Sébastien Sereni ◽  
Zelealem B. Yilma
Keyword(s):  
Lower Bound ◽  
Constant Factor ◽  
Petersen Graph ◽  
Cubic Graph ◽  
Permutation Graph ◽  
Permutation Graphs

A permutation graph is a cubic graph admitting a 1-factor $M$ whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if $e$ is an edge of $M$ such that every 4-cycle containing an edge of $M$ contains $e$, then $e$ is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of $M$ is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen subdivisions in a permutation graph with no 4-cycles, and give a construction showing that this lower bound is tight up to a constant factor.

scholarly journals Perfect Matching Covers of Cubic Graphs of Oddness 2

10.37236/7175 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Wuyang Sun ◽  
Fan Wang
Keyword(s):  
Perfect Matching ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Cubic Graph

A perfect matching cover of a graph $G$ is a set of perfect matchings of $G$ such that each edge of $G$ is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer $c$ does exist such that every bridgeless cubic graph has a perfect matching cover of order at most $c$. In this paper, we show that a bridgeless cubic graph $G$ has a perfect matching cover of order at most 11 if $G$ has a 2-factor in which the number of odd circuits is 2.

scholarly journals Covering a cubic graph by 5 perfect matchings

2017 ◽  
Vol 340 (8) ◽  
pp. 1889-1896
Author(s):  
Wuyang Sun
Keyword(s):  
Perfect Matchings ◽  
Cubic Graph

scholarly journals Treelike Snarks

10.37236/6008 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Marién Abreu ◽  
Tomáš Kaiser ◽  
Domenico Labbate ◽  
Giuseppe Mazzuoccolo
Keyword(s):  
Infinite Family ◽  
Double Cover ◽  
Perfect Matchings ◽  
Computer Assisted ◽  
Flow Number ◽  
Circular Flow

We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family  of such snarks, generalising an example provided by Hägglund. We  construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number $\phi_C (G)\ge5$ and admit a 5-cycle double cover.

scholarly journals Small Snarks with Large Oddness

10.37236/3969 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Robert Lukoťka ◽  
Edita Máčajová ◽  
Ján Mazák ◽  
Martin Škoviera
Keyword(s):  
Infinite Family ◽  
Upper Bounds ◽  
Petersen Graph ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Number

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\, \omega(G)$ vertices, and for each integer $k$ with $2\le k\le 6$ we construct an infinite family of cubic graphs with cyclic connectivity $k$ and small oddness ratio $|V(G)|/\omega(G)$. In particular, for cyclic connectivity $2$, $4$, $5$, and $6$ we improve the upper bounds on the oddness ratio of snarks to $7.5$, $13$, $25$, and $99$ from the known values $9$, $15$, $76$, and $118$, respectively. In addition, we construct a cyclically $4$-connected snark of girth $5$ with oddness $4$ on $44$ vertices, improving the best previous value of $46$. Corrigendum added March 19, 2018.

scholarly journals Covering a cubic graph with perfect matchings

2013 ◽  
Vol 313 (20) ◽  
pp. 2292-2296
Author(s):  
G. Mazzuoccolo
Keyword(s):  
Perfect Matchings ◽  
Cubic Graph
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