We prove that the chromatic number of any {3K1,K1∪K4}-free graph is at most a factor 28/15 times its clique number. In order to prove this result we prove that any connected subcubic triangle-free graph G on n vertices has a matching of size at least (n− 1)/3, and we characterise the extremal graphs.

A graph is prismatic if for every triangle T , every vertex not in T has exactly one neighbour in T . In this paper and the next in this series, we prove a structure theorem describing all prismatic graphs. This breaks into two cases depending whether the graph is 3-colourable or not, and in this paper we handle the 3-colourable case. (Indeed we handle a slight generalization of being 3-coloura...

‎let $g$ be a finite group‎. ‎in [ghasemabadi et al.‎, ‎characterizations of the simple group ${}^2d_n(3)$ by prime graph‎ ‎and spectrum‎, ‎monatsh math.‎, ‎2011] it is‎ ‎proved that if $n$ is odd‎, ‎then ${}^2d _n(3)$ is recognizable by‎ ‎prime graph and also by element orders‎. ‎in this paper we prove‎ ‎that if $n$ is even‎, ‎then $d={}^2d_{n}(3)$ is quasirecognizable by‎ ‎prime graph‎, ‎i.e‎...

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) Hamiltonian s-path, (2) a Hamiltonian st–path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containing...

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containi...

In this paper we study the chromatic number for graphs with forbidden induced subgraphs. We focus our interest on graph classes (de2ned in terms of forbidden induced subgraphs) for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can be determined also in polynomial time. Note that the 3-colourability decision problem is a well-known NP-co...

The Erdős-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n, where δ(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty b...

4 A 3-connected graph is 3-connected if it has no 3-separation that separates a “large” fan or K3,n 5 from the rest of the graph. It is proved in this paper that, except for K4, every 3 -connected graph has 6 a 3-connected proper minor that is at most two edges away from the original graph. This result is used 7 to characterize Q-minor-free graphs, where Q is obtained from the Cube by contracti...

A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a g...

In this paper, we propose a scheme to identify the maximal fault-free substar-ring. This is the rst result to derive a reconnguration scheme with high processor utilization in the faulty n-star graph. The maximal fault-free substar-ring is connected by a ring of fault-free virtual substars with dilation 3 and maximal length of the ring is n(n ? 1)(n ? 2). Our proposed scheme can tolerate n ? 3 ...