889 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
Approximating the Minimum Equivalent Digraph
The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994
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
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
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