1,783 research outputs found

    Cyclically five-connected cubic graphs

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    A cubic graph GG is cyclically 5-connected if GG is simple, 3-connected, has at least 10 vertices and for every set FF of edges of size at most four, at most one component of G\FG\backslash F contains circuits. We prove that if GG and HH are cyclically 5-connected cubic graphs and HH topologically contains GG, then either GG and HH are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph GG' such that HH topologically contains GG' and GG' is obtained from GG in one of the following two ways. Either GG' is obtained from GG by subdividing two distinct edges of GG and joining the two new vertices by an edge, or GG' is obtained from GG by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from GG in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of HH we are able to eliminate the second construction. We also prove versions of both of these results when GG is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case GG' is required to be almost cyclically 5-connected and to have fewer circuits of length four than GG. In particular, if GG has at most one circuit of length four, then GG' is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs GG' are more complicated.Comment: 47 pages, 5 figures. Revised according to referee's comments. To appear in J. Combin. Theory Ser.

    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

    A superlinear bound on the number of perfect matchings in cubic bridgeless graphs

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    Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde

    Excluded minors in cubic graphs

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    Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either G\v is planar for some vertex v, or G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.Comment: 62 pages, 17 figure

    Cubic graphs with large circumference deficit

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    The circumference c(G)c(G) of a graph GG is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 44-, 55- and 66-edge-connected cubic graphs with circumference ratio c(G)/V(G)c(G)/|V(G)| bounded from above by 0.8760.876, 0.9600.960 and 0.9900.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 44-edge-connected cubic graph is at least 0.750.75. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544]

    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

    Covering cubic graphs with matchings of large size

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    Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case m=n, that is a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some details the case m=n-1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has excessive [n-1]-index at most 4. Finally, we discuss the relation between excessive [n-1]-index and some other graph parameters as oddness and circumference.Comment: 11 pages, 5 figure

    Generation and Properties of Snarks

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    For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on n36n\leq 36 vertices. Previously lists up to n=28n=28 vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. 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.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201
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