9 research outputs found
Generalization of matching extensions in graphs (II)
Proposed as a general framework, Liu and Yu(Discrete Math. 231 (2001)
311-320) introduced -graphs to unify the concepts of deficiency of
matchings, -factor-criticality and -extendability. Let be a graph and
let and be non-negative integers such that and
is even. If when deleting any vertices from , the remaining
subgraph of contains a -matching and each such - matching can be
extended to a defect- matching in , then is called an
-graph. In \cite{Liu}, the recursive relations for distinct parameters
and were presented and the impact of adding or deleting an edge also
was discussed for the case . In this paper, we continue the study begun
in \cite{Liu} and obtain new recursive results for -graphs in the
general case .Comment: 12 page
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
On 1-factors and matching extension
AbstractWe prove the following: (1) Let G be a graph with a 1-factor and let F be an arbitrary 1-factor of G. If Gβ§Ή{a,b} is k-extendable for each abβF, then G is k-extendable. (2) Let G be a graph and let M be an arbitrary maximal matching of G. If Gβ§Ή{a,b} is k-factor-critical for each abβM, then G is k-factor-critical
The reducibility of optimal 1-planar graphs with respect to the lexicographic product
A graph is called 1-planar if it can be drawn on the plane (or on the sphere)
such that each edge is crossed at most once. A 1-planar graph is called
optimal if it satisfies . If and are graphs, then
the lexicographic product has vertex set the Cartesian product
and edge set . A graph is
called reducible if it can be expressed as the lexicographic product of two
smaller non-trivial graphs. In this paper, we prove that an optimal 1-planar
graph is reducible if and only if is isomorphic to the complete
multipartite graph . As a corollary, we prove that every reducible
1-planar graph with vertices has at most edges for or . We also prove that this bound is tight for infinitely many values of .
Additionally, we give two necessary conditions for a graph to be
1-planar.Comment: 23 pages, 14 fugure
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
The maximum matching extendability and factor-criticality of 1-planar graphs
A graph is - if it can be drawn in the plane so that each edge is
crossed by at most one other edge. Moreover, a 1-planar graph is
if it satisfies . J. Fujisawa et al. [16] first considered
matching extension of optimal 1-planar graphs, obtained that each optimal
1-planar graph of even order is 1-extendable and characterized 2-extendable
optimal 1-planar graphs and 3-matchings extendable to perfect matchings as
well. In this short paper, we prove that no optimal -planar graph is
3-extendable. Further we mainly obtain that no 1-planar graph is 5-extendable
by the discharge method and also construct a 4-extendable 1-planar graph.
Finally we get that no 1-planar graph is 7-factor-critical and no optimal
1-planar graph is 6-factor-critical.Comment: 13 pages, 8 figure