189 research outputs found

### On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

### A general framework for coloring problems: old results, new results, and open problems

In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants

### A note on K4-closures in Hamiltonian graph theory

Let G=(V, E) be a 2-connected graph. We call two vertices u and v of G a K4-pair if u and v are the vertices of degree two of an induced subgraph of G which is isomorphic to K4 minus an edge. Let x and y be the common neighbors of a K4-pair u, v in an induced K4−e. We prove the following result: If N(x)N(y)N(u)N(v){u,v}, then G is hamiltonian if and only if G+uv is h amiltonian. As a consequence, a claw-free graph G is hamiltonian if and only if G+uv is hamiltonian, where u,v is a K4-pair. Based on these results we define socalled K4-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K4-closures

### 3-Connected line graphs of triangular graphs are panconnected and 1-hamiltonian

A graph is k-triangular if each edge is in at least k triangles. Triangular is a synonym for 1-triangular. It is shown that the line graph of a triangular graph of order at least 4 is panconnected if and only if it is 3-connected. Furthermore, the line graph of a k-triangular graph is k-hamiltonian if and only if it is (k + 2)-connected (k ≥ 1). These results generalize work of Clark and Wormald and of Lesniak-Foster. Related results are due to Oberly and Sumner and to Kanetkar and Rao

### A closure concept based on neighborhood unions of independent triples

The well-known closure concept of Bondy and Chvatal is based on degree-sums of pairs of nonadjacent (independent) vertices. We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples. We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal

### On minimum degree conditions for supereulerian graphs

A graph is called supereulerian if it has a spanning closed trail. Let $G$ be a 2-edge-connected graph of order $n$ such that each minimal edge cut $E \subseteq E (G)$ with $|E| \le 3$ satisfies the property that each component of $G-E$ has order at least $(n-2)/5$. We prove that either $G$ is supereulerian or $G$ belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree $\delta\ge 4$: If $G$ is a 2-edge-connected graph of order $n$ with $\delta (G)\ge 4$ such that for every edge $xy\in E (G)$ , we have $\max \{d(x),d(y)\} \ge (n-7)/5$, then either $G$ is supereulerian or $G$ belongs to one of two classes of exceptional graphs. We show that the condition $\delta(G)\ge 4$ cannot be relaxed

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