How is a Chordal Graph like a Supersolvable Binary Matroid?

Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.Comment: 10 pages, 3 figures, to appear in Discrete Mathematic

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

A representation for the modules of a graph and applications

We describe a simple representation for the modules of a graph C. We show that the modules of C are in one-to-one correspondence with the ideaIs of certain posets. These posets are characterizaded and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate alI modules of C, (ii) count the number of modules of C, (iii) find a maximal module satisfying some hereditary property of C and (iv) find a connected non-trivial module of C

An algorithm for finding homogeneous pairs

AbstractA homogeneous pair in a graph G = (V, E) is a pair Q1, Q2 of disjoint sets of vertices in this graph such that every vertex of V (Q1 ∪ Q2) is adjacent either to all vertices of Q1 or to none of the vertices of Q1 and is adjacent either to all vertices of Q2 or to none of the vertices of Q2. Also ¦Q1¦ ⩾ 2 or ¦Q2¦⩾ 2 and ¦V (Q1 ∪ Q2)¦ ⩾ 2. In this paper we present an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one

New Results on Edge-coloring and Total-coloring of Split Graphs

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most $t$. Given a graph $G$, determining the smallest $t$ for which $G$ is $t$-admissible, i.e. the stretch index of $G$ denoted by $\sigma(G)$, is the goal of the $t$-admissibility problem. Split graphs are $3$-admissible and can be partitioned into three subclasses: split graphs with $\sigma=1, 2$ or $3$. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with $\Delta$ or $\Delta+1$ colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, i.e., a total coloring of $G$, it is conjectured that any graph can be total colored with $\Delta+1$ or $\Delta+2$ colors, and thus can be classified as Type 1 or Type 2. These both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with $\sigma=2$. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.Comment: 20 pages, 5 figure

Chordal (2,1) - graphs

A graph is said to be (k, l) if itsif its vertex set can be partitioned into k independent sets and l cliques. The class of (k,l) graphsappears as a natural generalization of split graphs. In this paper, we describe a characterization leads to a O (nm) recognition algorithm, where n and m are the numbers of vertices and edges of the input graph, respectively