A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free 2-factor. The toughness condition of result sharp. Moreover, for any $\varepsilon>0$ there exists $(2-\varepsilon)$-tough $2P_5$-free without This implies the $P_4 P_{10}$ best possible forbidden in sense.

We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H , testing whether a graph is H-free can be done in a constant...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

Let H be a graph. A graph G is said to be H -free if it contains no subgraph isomorphic to H . A graph G is said to be an H -saturated subgraph of a graph K if G is an H -free subgraph of K with the property that for any edge e∈E(K) \ E(G); G ∪ {e} is not H -free. We present some general results on Ks; t-saturated subgraphs of the complete bipartite graph Km;n and study the problem of 3nding, f...

Let G be a 2-connected claw-free graph. We show that • if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs, • if the minimum degree δ(G) ≥ 3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning euleri...

We show that every connected induced subgraph of a graph G is dominated by an split if and only C-free, where C set six graphs which includes P7 C7, each containing P5. A similar characterization shown for the class are complete graph. Motivated these results, we study structural descriptions some classes (P7, C7)-free graphs. In particular, give C4, gem)-free diamond)-free Using satisfies χ(G)...

A pair of vertices of a graph is called an even pair if every chordless path between them has an even number of edges. A graph is minimally even pair free if it is not a clique, contains no even pair, but every proper induced subgraph either contains an even pair or is a clique. Hougardy (European J. Combin. 16 (1995) 17–21) conjectured that a minimally even pair free graph is either an odd cyc...

Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, inclu...

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...

In this paper, we will consider the free probability on the free group factor L(F N), in terms of Graph W *-probability, where F k is the free group with k-generators. The main result of this paper is to reformulate the moment series and the R-transform of the operator T 0 = N j=1 (x j) , where x j 's are free semicircular elements, j = 1, ..., N, generating L(F N), by using the Graph W *-proba...