1,268 research outputs found

### 4-Factor-criticality of vertex-transitive graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the
same parity as $n$, if the removal of any set of $p$ 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

### 2-Resonant fullerenes

A fullerene graph $F$ is a planar cubic graph with exactly 12 pentagonal
faces and other hexagonal faces. A set $\mathcal{H}$ of disjoint hexagons of
$F$ is called a resonant pattern (or sextet pattern) if $F$ has a perfect
matching $M$ such that every hexagon in $\mathcal{H}$ is $M$-alternating. $F$
is said to be $k$-resonant if any $i$ ($0\leq i\leq k$) disjoint hexagons of
$F$ form a resonant pattern. It was known that each fullerene graph is
1-resonant and all 3-resonant fullerenes are only the nine graphs. In this
paper, we show that the fullerene graphs which do not contain the subgraph $L$
or $R$ as illustrated in Fig. 1 are 2-resonant except for the specific eleven
graphs. This result implies that each IPR fullerene is 2-resonant.Comment: 34 pages, 25 figure

### 3-Factor-criticality of vertex-transitive graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the
same parity as $n$, if the removal of any set of $p$ 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

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