Let G be a graph. The vertex edge arboricity of G denoted by a G a1 G is the minimum number of subsets into which the vertex edge set of G can be partitioned so that each subset induces an acyclic subgraph. Let d be a graphical sequence and let R d be the class of realizations of d. We prove that if π ∈ {a, a1}, then there exist integers x π and y π such that d has a realization G with π G z if...

We introduce the concept of variable degeneracy of a graph extending that of k-degeneracy. This makes it possible to give a common generalization of the point partition numbers and the list chromatic number. In particular, the list point arboricity of a graph is considered. We extend Brooks’ and Gallai’s theorems in terms of covering the vertices of a graph by disjoint induced subgraphs G1; : :...

A linear forest-factor F of a graph G is a spanning subgraph of G whose components are paths. A linear forest-decomposition of G is a collection :F = {F1, ••• , Fk } of linear forest-factors of G such that the edge set E (G) of G is the disjoint union of E (F1), •.• , E (Fk)' The linear ar borici ty la( G) of G is the minimum cardinality of a linear forest-decomposition of G. In this paper we e...

In an ordinary edge-coloring of a graph each color appears at each vertex at most once. An f -coloring is a generalized edge-coloring in which each color appears at each vertex v at most f(v) times where f(v) is a positive integer assigned to v. This paper gives efficient sequential and parallel algorithms to find ordinary edge-colorings and f -colorings for various classes of graphs such as bi...

A linear forest is a forest in which every component is a path. It is known that the set of vertices V (G) of any outerplanar graph G can be partitioned into two disjoint subsets V1, V2 such that induced subgraphs 〈V1〉 and 〈V2〉 are linear forests (we say G has an (LF ,LF)partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of i...

The vertex [edge] arboricity a(G) [a1(G)] of a graph G is the minimum number of subsets into which V (G) [E(G)] can be partitioned so that each subset induces an acyclic subgraph. Let G(m,n) be the class of connected simple graphs of order n and size m and let π ∈ {a, a1}. In this paper we determine π(m,n) := {π(G) : G ∈ G(m,n)} for integers m,n such that n− 1 ≤ m ≤ (n 2 ) .

Let G = (V,E) be an n-vertex graph and Md a d-vertex graph, for some constant d. Is Md a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n(d−2)/d/ log n) communication rounds is presented. For the special case that Md is a triangle, we pr...

We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H . It is known that Connected Ver...