In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an n-vertex graph with O(n) edges and (n/ log n) maximal complete bipartite subgraphs Kk,`. ∗Work supported in part by NSF grant CCR-9258355.

We prove that every triangle-free planar graph of order n and size m has an induced linear forest with at least 9n−2m 11 vertices, and thus at least 5n+8 11 vertices. Furthermore, we show that there are triangle-free planar graphs on n vertices whose largest induced linear forest has order ⌈n2 ⌉+ 1.

Let D be a subset of the positive integers. The distance graph G(Z,D) has all integers as its vertices and two vertices x and y are adjacent if and only if |x − y| ∈ D, where the set D is called distance set. The vertex arboricity va(G) of a graph G is the minimum number of subsets into which vertex set V(G) can be partitioned so that each subset induces an acyclic subgraph. In this paper, the ...

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edge set E(G) of G. The vertex linear arboricity vla(G) of a graph G is the minimum number of subsets into which the vertex set V (G) can be partitioned so that every subset induces a linear forest. The Schrijver graph SG(n, k) is the graph whose vertex set consists of all 2-stable k-subsets of ...

Acyclic-coloring of a graph G = (V; E) is a partitioning of V , such that the induced subgraph of each partition is acyclic. The minimum number of such partitions of V is deened as the vertex arboricity of G. An O(n) algorithm (n = jV j) for acyclic-coloring of planar graphs with 3 colors is presented. Next, an O(n 2) heuristic is proposed which produces a valid acyclic-2-coloring of a planar g...

Acyclic-coloring of a graph G = (V;E) is a partitioning of V , such that the induced subgraph of each partition is acyclic. The minimumnumber of such partitions of V is de ned as the vertex arboricity of G. An O(n) algorithm (n = jV j) for acyclic-coloring of planar graphs with 3 colors is presented. Next, an O(n) heuristic is proposed which produces a valid acyclic-2-coloring of a planar graph...