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### The number of cycles in 2-factors of cubic graphs

Moshe Rosenfeld
1990 Discrete Mathematics
The functions 5 can be viewed as the average length of cycles in 2-factors of the extremal graphs. f0 = lim inf IG, The main results Let G and H be two disjoint cubic 3-connected graphs.  ...  We denote by '9 the family of all cubic 3-connected simple graphs and by sr, the subset of planar graphs in 5% The number of vertices in a graph G is denoted by ICI.  ...  The graph G in Fig. 8  ...

### Klein Group And Four Color Theorem [article]

Sergey Kurapov
2010 arXiv   pre-print
In this work methods of construction of cubic graphs are analyzed and a theorem of existence of a colored disc traversing each pair of linked edges belonging to an elementary cycle of a planar cubic graph  ...  The number of edges in a cubic graph is determined as: m = 3n / 2 (1) Hence, the number of edges in such a graph is always a multiple of three.  ...  Since the number of edges m is integer, therefore, the number of vertices n in an isomorphic cubic graph is even.  ...

### Switching 3-edge-colorings of cubic graphs [article]

Jan Goedgebeur, Patric R. J. Östergård
2021 arXiv   pre-print
Families of cubic graphs of orders 4n+2 and 4n+4 with 2^n edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs with more edge-Kempe equivalence classes.  ...  The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs.  ...  The observation at the end of Section 2 dates back to discussions with Petteri Kaski after the publication of [24] .  ...

### Generation and properties of snarks

Gunnar Brinkmann, Jan Goedgebeur, Jonas Hägglund, Klas Markström
2013 Journal of combinatorial theory. Series B (Print)
In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.  ...  In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures.  ...  Jan Goedgebeur is supported by a PhD grant of the Research Foundation of Flanders (FWO). Jonas Hägglund is supported by the National Graduate School in Scientific Computing (NGSSC).  ...

### On S-packing edge-colorings of cubic graphs

Nicolas Gastineau, Olivier Togni
2019 Discrete Applied Mathematics
Given a non-decreasing sequence S = (s 1 , s 2 , . . . , s k ) of positive integers, an Spacking edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 , X 2 , . . . , X k }  ...  Among other results, we prove that cubic graphs having a 2-factor are (1, 1, 1, 3 , 3)-packing edge-colorable, (1, 1, 1, 4, 4, 4, 4, 4) -packing edgecolorable and (1, 1, 2, 2, 2, 2, 2)-packing edge-colorable  ...  For a cubic graph G having a 2-factor, the oddness of G is the minimum number of odd cycle among all 2-factors of G. According to Petersen's theorem, every bridgeless cubic graph has a 2-factor.  ...

### On S-packing edge-colorings of cubic graphs [article]

Nicolas Gastineau
2017 arXiv   pre-print
Given a non-decreasing sequence S = (s 1,s 2,. .. ,s k) of positive integers, an S-packing edge-coloring of a graph G is a partition of the edge set of G into k subsets X 1 ,X 2,. ..  ...  Among other results, we prove that cubic graphs having a 2-factor are (1,1,1,3,3)-packing edge-colorable, (1,1,1,4,4,4,4,4)-packing edge-colorable and (1,1,2,2,2,2,2)-packing edge-colorable.  ...  For a cubic graph G having a 2-factor, the oddness of G is the minimum number of odd cycle among all 2-factors of G. According to Petersen's theorem, every bridgeless cubic graph has a 2-factor.  ...

### Improved Approximations for Cubic and Cubic Bipartite TSP [article]

Anke van Zuylen
2016 arXiv   pre-print
For 2-connected cubic graphs, we show that the techniques of Moemke and Svensson (2011) can be combined with the techniques of Correa, Larre and Soto (2012), to obtain a tour of length at most (4/3-1/8754  ...  We show improved approximation guarantees for the traveling salesman problem on cubic graphs, and cubic bipartite graphs.  ...  for suggesting the simplified proof for the result in Section 3 for cubic non-bipartite graphs.  ...

### Improved Approximations for Cubic Bipartite and Cubic TSP [chapter]

Anke van Zuylen
2016 Lecture Notes in Computer Science
For 2-connected cubic graphs, we show that the techniques of Mömke and Svensson can be combined with the techniques of Correa, Larré and Soto, to obtain a tour of length at most (4/3 − 1/8754)n.  ...  We show improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs.  ...  for suggesting the simplified proof for the result in Section 3 for cubic non-bipartite graphs.  ...

### On snarks that are far from being 3-edge colorable [article]

Jonas Hägglund
2012 arXiv   pre-print
Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.  ...  In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices.  ...  The oddness of a bridgeless cubic graph G is defined as the minimum number of odd components in any 2-factor in G and is denoted by o(G).  ...

### Shortest cycle covers and cycle double covers with large 2-regular subgraphs

Jonas Hägglund, Klas Markström
2013 Journal of Combinatorics
In this paper, we show that many snarks have a shortest cycle cover of length 4 3 m + c for a constant c, where m is the number of edges in the graph, in agreement with the conjecture that all snarks have  ...  In particular, we prove that graphs with perfect matching index at most 4 have cycle covers of length 4 3 m and satisfy the (1, 2)covering conjecture of Zhang, and that graphs with large circumference  ...  Acknowledgments The authors would like to thank the referees for their constructive criticism and Lars-DanielÖhman for his comments on the manuscript.  ...

### Improved bounds for the shortness coefficient of cyclically 4-edge connected cubic graphs and snarks [article]

Klas Markström
2014 arXiv   pre-print
The graphs we construct are snarks so we get the same upper bound for the shortness coefficient of snarks, and we prove that the constructed graphs have an oddness growing linearly with the number of vertices  ...  We present a construction which shows that there is an infinite set of cyclically 4-edge connected cubic graphs on n vertices with no cycle longer than c_4 n for c_4=12/13, and at the same time prove that  ...  One measure of the degree of non-colourability for a cubic graph is its oddness, i.e. the minimum number of odd cycles in any 2-factor of the graph.  ...

### The Color Number of Cubic Graphs Having a Spanning Tree with a Bounded Number of Leaves

Analen A Malnegro, Gina Malacas, Kenta Ozeki
2021 Theory and Applications of Graphs
In this paper, we extend these observations by obtaining a bound for the color number of cubic graphs having a spanning tree with a bounded number of leaves.  ...  The color number c(G) of a cubic graph G is the minimum cardinality of a color class of a proper 4-edge-coloring of G.  ...  The oddness of a cubic graph G, denoted by ω(G), is the smallest number of odd cycles in a 2-factor of G, where a 2-factor is a spanning subgraph in which every vertex has degree 2.  ...

### Shortest cycle covers and cycle double covers with large 2-regular subgraphs [article]

Jonas Hägglund, Klas Markstrøm
2013 arXiv   pre-print
In this paper we show that many snarks have shortest cycle covers of length 4/3m+c for a constant c, where m is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest  ...  In particular we prove that graphs with perfect matching index at most 4 have cycle covers of length 4/3m and satisfy the (1,2)-covering conjecture of Zhang, and that graphs with large circumference have  ...  Short covers in graphs with Oddness 2 Recall that the oddness o(G) of a cubic graph G is the minimum number of odd cycles in any 2-factor of G.  ...

### Tutte's 5-flow conjecture for highly cyclically connected cubic graphs

Eckhard Steffen
2010 Discrete Mathematics
In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let ω be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph.  ...  We show that if a cubic graph G has no edge cut with fewer than 5/2ω - 1 edges that separates two odd cycles of a minimum 2-factor of G, then G has a nowhere-zero 5-flow.  ...  By Petersen's theorem, every bridgeless cubic graph G has a 2-factor and the oddness ω(G) is the minimum number of odd cycles in a 2-factor of G.  ...

### Weak oddness as an approximation of oddness and resistance in cubic graphs [article]

Robert Lukoťka, Ján Mazák
2016 arXiv   pre-print
We introduce weak oddness ω_ w, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor.  ...  For every bridgeless cubic graph G, ρ(G)<ω_ w(G)<ω(G), where ρ(G) denotes the resistance of G and ω(G) denotes the oddness of G, so this new measure is an approximation of both oddness and resistance.  ...  We would like to thank Barbora Candráková, Edita Máčajová, Eckhard Steffen, and Martin Škoviera for many fruitful discussions on topics related to even factors in cubic graphs.  ...
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