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We consider a generalization of edge intersection graphs of paths in a tree. Let P be a collection of nontrivial simple paths in a tree T. We define the k-edge (k 1) intersection graph k (P), whose vertices correspond to the members of P, and two vertices are joined by an edge if the corresponding members of P share k edges in T. An undirected graph G is… (More)

A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which… (More)

The subchromatic number χ S (G) of a graph G = (V, E) is the smallest order k of a partition {V 1 , V 2 ,. .. , V k } of the vertices V (G) such that the subgaph < V i > induced by each subset V i consisits of a disjoint union of complete subgraphs. By definition, χ S (G) ≤ χ(G), the chromatic number of G. This paper develops properties of this lower bound… (More)

Given two graphs G and H, let f (G,H) denote the minimum integer n such that in every coloring of the edges of K n , there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f (G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we… (More)

We obtain bounds for the coloring numbers of products of trees for three closely related types of colorings: acyclic, distance 2, and L(2, 1).

A chordal graph is the intersection graph of a family of subtrees of a tree, or, equivalently, it is the (edge-)intersection graph of leaf-generated subtrees of a full binary tree. In this paper, a generalization of chordal graphs from this viewpoint is studied: a graph G=(V; E) is representable if there is a family of subtrees {S v}v∈V of a binary tree,… (More)