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

Super cyclically edge connected graphs with two orbits of the same size

对于图$G$,如果$G -; F$是不连通的且至少有两个分支含有圈,则称$F$为图$G$的圈边割.如果图$G$有圈边割,则称其为圈可分的.最小圈边割的基数叫作圈边连通度.如果; 去除任何一个最小圈边割,总存在一分支为最小圈,则图$G$为超圈边连通的.设$G = \left( {{G_1},{G_2},\left(; {{V_1},{V_2}} \right)} \right)$为双轨道图,最小度$\delta \left( G \right) \ge; 4$,围长$g\left( G \right) \ge 6$且$\left| {{V_1}} \right| = \left| {{V_2}}; \right|$.假设${G_i}$是${k_i}$-正则的,${k_1} \le; {k_2}$且${{G_1}}$包含一个长度为$g$的圈,则$G$是超圈边连通的.For a graph $G$, an edge set $F$ is a cyclic edge-cut if ($G - F$) is; disconnected and at least two of its components contain cycles. If $G$; has a cyclic edge-cut, it is said to be cyclically separable. The cyclic; edge-connectivity is cardinality of a minimum cyclic edgecut of $G$. A; graph is super cyclically edge-connected if removal of any minimum; cyclic edge-cut makes a component a shortest cycle. Let $G = \left(; {{G_1},{G_2},\left( {{V_1},{V_2}} \right)} \right)$ be a doubleorbit; graph with minimum degree $\delta \left( G \right) \ge 4$, girth $g \ge; 6$ and $\left| {{V_1}} \right| = \left| {{V_2}} \right|$. Suppose; ${G_i}$ is ${k_i}$-regular, ${k_1} \le {k_2}$ and ${{G_1}}$ contains a; cycle of length $g$, then $G$ is super cyclically edge connected.国家自然科学基金资助项

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

Variational networks of cube-connected cycles are recursive cubes of rings

In this short note we show that the interconnection networks known as variational networks of cube-connected cycles form a sub-class of the recursive cubes of rings

Cycle density in infinite Ramanujan graphs

We introduce a technique using nonbacktracking random walk for estimating the spectral radius of simple random walk. This technique relates the density of nontrivial cycles in simple random walk to that in nonbacktracking random walk. We apply this to infinite Ramanujan graphs, which are regular graphs whose spectral radius equals that of the tree of the same degree. Kesten showed that the only infinite Ramanujan graphs that are Cayley graphs are trees. This result was extended to unimodular random rooted regular graphs by Ab\'{e}rt, Glasner and Vir\'{a}g. We show that an analogous result holds for all regular graphs: the frequency of times spent by simple random walk in a nontrivial cycle is a.s. 0 on every infinite Ramanujan graph. We also give quantitative versions of that result, which we apply to answer another question of Ab\'{e}rt, Glasner and Vir\'{a}g, showing that on an infinite Ramanujan graph, the probability that simple random walk encounters a short cycle tends to 0 a.s. as the time tends to infinity.Comment: Published at http://dx.doi.org/10.1214/14-AOP961 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org