A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are precisely the minimal forbidden induced circular-arc subgraphs for the class of Helly circular-arc graphs. We show that it is possible to find in linear time, in an...

We introduce the incidence game chromatic number which unifies the ideas of game chromatic number and incidence coloring number of an undirected graph. For kdegenerate graphs with maximum degree ∆, the upper bound 2∆ + 4k − 2 for the incidence game chromatic number is given. If ∆ ≥ 5k, we improve this bound to the value 2∆ + 3k − 1. We also determine the exact incidence game chromatic number of...

Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (e.g. the class of perfect graphs and the class of even-hole-free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced sub...

A wheel graph is a formed by connecting single vertex to all vertices of cycle. called wheel-free if it does not contain any as subgraph. In 2010, Nikiforov proposed Brualdi–Solheid–Turán type problem: what the maximum spectral radius order n that subgraphs particular kind. this paper, we study problem for graphs, and determine (signless Laplacian) n. Furthermore, characterize extremal graphs.

In Euler and Mahjoub (1991) it is proved that the triangle-free subgraph polytope of a graph noncontractible to K,\e is completely described by the trivial inequalities and the so-called triangle and odd wheel inequalities. In this paper we show that the system defined by those inequalities is TDI for a subclass of that class of graphs. As a consequence we obtain the following min-max relation:...

A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdős asserted that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a quest...

A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dmin and dmax, dmin ≤ dmax, such that each node u ∈ V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin ≤ dT (u, v) ≤ dmax, where dT (u, v) is the sum of the weights of the edges on the unique path PT (u, v) from u to v in T...

In a graph G, a k-insulated set S is a subset of the vertices of G such that every vertex in S is adjacent to at most k vertices in S, and every vertex outside S is adjacent to at least k + 1 vertices in S. The insulation sequence i0, i1, i2, . . . of a graph G is defined by setting ik equal to the maximum cardinality of a k-insulated set in G. We determine the insulation sequence for paths, cy...

Let G and H be graphs. The Ramsey number R(G, H) is the least integer such that for every graph F of order R(G, H), either F contains G or F contains H . Let Bn and Wn denote the book graph K2 +Kn and the wheel graph K1 + Cn−1, respectively. In 1978, Faudree, Rousseau and Sheehan computed R(C4, Bn) for n ≤ 8. In this paper, we compute R(C4, Bn) for 8 ≤ n ≤ 12 and R(C4, Wn) for 4 ≤ n ≤ 13. In pa...

In this paper we show that, if G is a Berge graph such that neither G nor its complement G contains certain induced subgraphs, named proper wheels and long prisms, then either G is a basic perfect graph (a bipartite graph, a line graph of a bipartite graph or the complement of such graphs) or it has a skew partition that cannot occur in a minimally imperfect graph. This structural result implie...