On clique-colouring of graphs with few P4's

Abstract Let G=(V,E) be a graph with n vertices. A clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A k-clique-colouring is a clique-colouring that uses k colours. The clique-chromatic number of a graph G is the minimum k such that G has a k-clique-colouring. In this paper we will use the primeval decomposition technique to find the clique-chromatic number and the clique-colouring of well known classes of graphs that in some local sense contain few P 4's. In particular we shall consider the classes of extended P 4-laden graphs, p-trees (graphs which contain exactly n−3 P 4's) and (q,q−3)-graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P 4's. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, P 4-reducible graphs, P 4-sparse graphs, extended P 4-reducible graphs, extended P 4-sparse graphs, P 4-extendible graphs, P 4-lite graphs, P 4-tidy graphs and P 4-laden graphs that are included in the class of extended P 4-laden graphs

On the classification problem for split graphs

Abstract The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs

On clique-colouring of graphs with few P4

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The induced path convexity, betweenness, and svelte graphs

AbstractThe induced path interval J(u,v) consists of the vertices on the induced paths between u and v in a connected graph G. Differences in properties with the geodesic interval are studied. Those graphs are characterized, in which the induced path intervals define a proper betweenness. The intersection of the induced path intervals between the pairs of a triple, in general, consists of a big chunk of vertices. The graphs, in which this intersection consists of at most one vertex, for each triple of vertices, are characterized by forbidden subgraphs

Well Covered Graphs with Few P4´s

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Tutte sets in graphs I: Maximal tutte sets and D-graphs

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency of G. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph G. Towards this end, we introduce a related graph D(G). We first show that the maximal Tutte sets in G are precisely the maximal independent sets in its D-graph D(G), and then continue with the study of D-graphs in their own right, and of iterated D-graphs. We show that G is isomorphic to a spanning subgraph of D(G), and characterize the graphs for which G ≅ D(G) and for which D(G) ≅ D2(G). Surprisingly, it turns out that for every graph G with a perfect matching, D3(G) ≅ D2(G). Finally, we characterize bipartite D-graphs and comment on the problem of characterizing D-graphs in general. © 2007 Wiley Periodicals, Inc