The diamond is the graph obtained by removing an edge from complete on 4 vertices. A (\(P_6\), diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or diamond. In this paper we show that chromatic number of G larger than maximum 6 and clique G. We do reducing problem imperfect graphs via Strong Perfect Graph Theorem, dividing into several cases, giving proper colouri...

Given two graphs H 1 and 2 , a graph is ( ) -free if it contains no induced subgraph isomorphic to or . For positive integer t, P t the chordless path on vertices. A paraglider that consists of chorless cycle C 4 plus vertex adjacent three vertices In this paper, we study structure 5 paraglider)-free graphs, show every such G satisfies ? ? ? 3 ? ? where are chromatic number clique G, respective...

For an integer , a k-path vertex cover of graph is set that shares with every path subgraph order k in G. The minimum cardinality denoted by . We give estimates—mostly upper bounds—on terms various parameters, including degrees and the number vertices edges. problem also considered on chordal graphs planar graphs.

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colours needed for a rainbow edge (vertex) colouring of G. In this paper we propose a very simple approach to studying rainbow connectivity in...

A graph is shown to be distance-hereditary if and only if no induced subgraph of order five or more has a unique hamiltonian cycle; this is also equivalent to every induced subgraph of order five or more having an even number of hamiltonian cycles. Restricting the induced subgraphs to those of odd order five or more gives two similar characterizations of parity graphs. The close relationship be...