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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)
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)
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)