4,419 research outputs found
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
4-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 well-known factor-critical graphs and bicritical graphs,
respectively. It is known that if a connected vertex-transitive graph has odd
order, then it is factor-critical, otherwise it is elementary bipartite or
bicritical. In this paper, we show that a connected vertex-transitive
non-bipartite graph of even order at least 6 is 4-factor-critical if and only
if its degree is at least 5. This result implies that each connected
non-bipartite Cayley graphs of even order and degree at least 5 is
2-extendable.Comment: 34 pages, 3 figure
Extendable self-avoiding walks
The connective constant mu of a graph is the exponential growth rate of the
number of n-step self-avoiding walks starting at a given vertex. A
self-avoiding walk is said to be forward (respectively, backward) extendable if
it may be extended forwards (respectively, backwards) to a singly infinite
self-avoiding walk. It is called doubly extendable if it may be extended in
both directions simultaneously to a doubly infinite self-avoiding walk. We
prove that the connective constants for forward, backward, and doubly
extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB,
exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite,
strongly connected, quasi-transitive directed graph. The proofs rely on a 1967
result of Furstenberg on dimension, and involve two different arguments
depending on whether or not the graph is unimodular.Comment: Accepted versio
Maximizing the minimum and maximum forcing numbers of perfect matchings of graphs
Let be a simple graph with vertices and a perfect matching. The
forcing number of a perfect matching of is the smallest
cardinality of a subset of that is contained in no other perfect matching
of . Among all perfect matchings of , the minimum and maximum values
of are called the minimum and maximum forcing numbers of , denoted
by and , respectively. Then . Che and Chen
(2011) proposed an open problem: how to characterize the graphs with
. Later they showed that for bipartite graphs , if and
only if is complete bipartite graph . In this paper, we solve the
problem for general graphs and obtain that if and only if is a
complete multipartite graph or ( with arbitrary additional
edges in the same partite set). For a larger class of graphs with
we show that is -connected and a brick (3-connected and
bicritical graph) except for . In particular, we prove that the
forcing spectrum of each such graph is continued by matching 2-switches and
the minimum forcing numbers of all such graphs form an integer interval
from to
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