459 research outputs found

    An exploration of two infinite families of snarks

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    Thesis (M.S.) University of Alaska Fairbanks, 2019In this paper, we generalize a single example of a snark that admits a drawing with even rotational symmetry into two infinite families using a voltage graph construction techniques derived from cyclic Pseudo-Loupekine snarks. We expose an enforced chirality in coloring the underlying 5-pole that generated the known example, and use this fact to show that the infinite families are in fact snarks. We explore the construction of these families in terms of the blowup construction. We show that a graph in either family with rotational symmetry of order m has automorphism group of order m2m⁺¹. The oddness of graphs in both families is determined exactly, and shown to increase linearly with the order of rotational symmetry.Chapter 1: Introduction -- 1.1 General Graph Theory -- Chapter 2: Introduction to Snarks -- 2.1 History -- 2.2 Motivation -- 2.3 Loupekine Snarks and k-poles -- 2.4 Conditions on Triviality -- Chapter 3: The Construction of Two Families of Snarks -- 3.1 Voltage Graphs and Lifts -- 3.2 The Family of Snarks, Fm -- 3.3 A Second Family of Snarks, Rm -- Chapter 4: Results -- 4.1 Proof that the graphs Fm and Rm are Snarks -- 4.2 Interpreting Fm and Rm as Blowup Graphs -- 4.3 Automorphism Group -- 4.4 Oddness -- Chapter 5: Conclusions and Open Questions -- References

    The Cost of Perfection for Matchings in Graphs

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    Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs

    On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings

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    The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F\cal F of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs were known. It turns out that the family F\cal F also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with mm edges has length at least 43m\tfrac43m, and we show that this inequality is strict for graphs of F\cal F. We also construct the first known snark with no cycle cover of length less than 43m+2\tfrac43m+2.Comment: 17 pages, 8 figure

    Normal 6-edge-colorings of some bridgeless cubic graphs

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    In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal kk-edge-coloring of a cubic graph is an edge-coloring with kk colors such that each edge of the graph is normal. We denote by χN(G)\chi'_{N}(G) the smallest kk, for which GG admits a normal kk-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving χN(G)5\chi'_{N}(G)\leq 5 for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal 77-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal 66-edge-coloring. Finally, we show that any bridgeless cubic graph GG admits a 66-edge-coloring such that at least 79E\frac{7}{9}\cdot |E| edges of GG are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with arXiv:1804.0944

    Petersen cores and the oddness of cubic graphs

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    Let GG be a bridgeless cubic graph. Consider a list of kk 1-factors of GG. Let EiE_i be the set of edges contained in precisely ii members of the kk 1-factors. Let μk(G)\mu_k(G) be the smallest E0|E_0| over all lists of kk 1-factors of GG. If GG is not 3-edge-colorable, then μ3(G)3\mu_3(G) \geq 3. In [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78(3) (2015) 195-206] it is shown that if μ3(G)0\mu_3(G) \not = 0, then 2μ3(G)2 \mu_3(G) is an upper bound for the girth of GG. We show that μ3(G)\mu_3(G) bounds the oddness ω(G)\omega(G) of GG as well. We prove that ω(G)23μ3(G)\omega(G)\leq \frac{2}{3}\mu_3(G). If μ3(G)=23μ3(G)\mu_3(G) = \frac{2}{3} \mu_3(G), then every μ3(G)\mu_3(G)-core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4-edge-connected cubic graph GG with ω(G)=23μ3(G)\omega(G) = \frac{2}{3}\mu_3(G). On the other hand, the difference between ω(G)\omega(G) and 23μ3(G)\frac{2}{3}\mu_3(G) can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer k3k\geq 3, there exists a bridgeless cubic graph GG such that μ3(G)=k\mu_3(G)=k.Comment: 13 pages, 9 figure
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