4,430 research outputs found

    Short Cycles Connectivity

    Full text link
    Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is an equivalence relation on the set of vertices, while the edge short cycles connectivity components determine an equivalence relation on the set of edges. Efficient algorithms for determining equivalence classes are presented. Short cycles connectivity can be extended to directed graphs (cyclic and transitive connectivity). For further generalization we can also consider connectivity by small cliques or other families of graphs

    3-Factor-criticality of vertex-transitive graphs

    Full text link
    A graph of order nn is pp-factor-critical, where pp is an integer of the same parity as nn, if the removal of any set of pp vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if and only if it is not a cycle.Comment: 15 pages, 3 figure

    Edge-transitivity of Cayley graphs generated by transpositions

    Full text link
    Let SS be a set of transpositions generating the symmetric group SnS_n. The transposition graph of SS is defined to be the graph with vertex set {1,,n}\{1,\ldots,n\}, and with vertices ii and jj being adjacent in T(S)T(S) whenever (i,j)S(i,j) \in S. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph T(S)T(S) is edge-transitive if and only if the Cayley graph Cay(Sn,S)Cay(S_n,S) is edge-transitive

    Notes on the connectivity of Cayley coset digraphs

    Full text link
    Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

    Hamilton cycles in dense vertex-transitive graphs

    Get PDF
    A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series
    corecore