A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic toK1,3. The kth power of a graphG, denoted byGk , is the graph with vertex set V (G) in which two vertices are adjacent if and only if the distance between them in G is at most k. I...

In 1968, Erdös and Lovász conjectured that for every graph G and all integers s, t ≥ 2 such that s + t − 1 = χ(G) > ω(G), there exists a partition (S, T ) of the vertex set of G such that χ(G|S) ≥ s and χ(G|T ) ≥ t. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stabili...

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominable graph (G is etd). In this paper, we prove NP-completeness of the etd decision problem on the class of planar bipartite graphs of maximum degree 3. Furthermore, we give an effic...

A graph G is locally connected if the subgraph induced by the neighbourhood of each vertex is connected. We prove that a locally connected graph G of order p 3, containing no induced subgraph isomorphic to K 1;3 , is Hamilton-connected if and only if G is 3-connected.

A graph G is list point k-arborable if, whenever we are given a k-list assignment L(v) of colors for each vertex v ∈ V(G), we can choose a color c(v) ∈ L(v) for each vertex v so that each color class induces an acyclic subgraph of G, and is equitable list point k-arborable if G is list point k-arborable and each color appears on at most ⌈|V(G)|/k⌉ vertices of G. In this paper, we conjecture tha...

In this paper, we study graphs with integer matching polynomial roots. We characterize all traceable graphs whose all matching polynomial roots are integer. We show that there is no k-regular claw-free graph (k ≥ 2), whose all matching polynomial roots are integer. Also we conjecture that apart from K7 \ (E(C3) ∪E(C4)), there is no matching integral k-regular graph (k ≥ 2). 2010 Mathematics Sub...

Given a graph G, its partially square graph G∗ is a graph obtained by adding an edge (u, v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition NG(x) ⊆ NG[u] ∪ NG[v], where NG[x] = NG(x) ∪ {x}. In the case where G is a claw-free graph, G∗ is equal to G. We define σ◦ t = min{ ∑ x∈S dG(x) : S is an independent set in G ...

We study the class of independence complexes of claw-free graphs. The main theorem give good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is (cn/d + ε)–connected, where c = 2/3. This can be compared with...

Seymour conjectured for a fixed integer k ≥ 2 that if G is a graph of order n with δ(G) ≥ kn/(k + 1), then G contains the kth power Ck n of a Hamiltonian cycle Cn of G, and this minimum degree condition is sharp. Earlier the k = 2 case was conjectured by Pósa. This was verified by Komlós et al. [4]. For s ≥ 3, a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. S...