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

Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results