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Short Cycles Connectivity
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
A graph of order is -factor-critical, where is an integer of the
same parity as , if the removal of any set of 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
Let be a set of transpositions generating the symmetric group . The
transposition graph of is defined to be the graph with vertex set
, and with vertices and being adjacent in
whenever . 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 is
edge-transitive if and only if the Cayley graph is
edge-transitive
Notes on the connectivity of Cayley coset digraphs
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
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
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