In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

The k-Coloring problem is to test whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. If a graph G does not contain a graph H as an induced subgraph, then G is called H-free. For any fixed graph H on at most 6 vertices, it is known that 3-Coloring is polynomial-time solvable on H-free graphs whenever H is a linear forest and NP-complet...

let z2 = {0, 1} and g = (v ,e) be a graph. a labeling f : v → z2 induces an edge labeling f* : e →z2 defined by f*(uv) = f(u).f (v). for i ε z2 let vf (i) = v(i) = card{v ε v : f(v) = i} and ef (i) = e(i) = {e ε e : f*(e) = i}. a labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. the vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. in this paper ...

It is proved that the linear arboricity of every 1-planar graph with maximum degree ∆ > 33 is ⌈∆/2⌉.

An induced matching in a graph G = (V,E) is a matching M such that (V,M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj, Pelsmajer, Schaefer and Xia [10] studied i...

Let H be a graph on h vertices, and let G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n=h vertex disjoint copies of H. The fractional arboricity of H is a(H) = maxf jE 0 j jV 0 j?1 g, where the maximum is taken over all subgraphs (V 0 ; E 0) of H with jV 0 j > 1. Let (H) denote the minimum degree of a vertex of H. It is shown that if (H) < a(H) then n ?1...

A d-track interval is a union of d intervals, one each from d parallel lines. The intersection graphs of d-track intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The track number for K m;n is determined by proving that the arboricity of K m;n equals its \caterpillar arbori...

In this paper we present a modification of a technique by Chiba and Nishizeki [Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J. Comput. 14(1), pp. 210–223 (1985)]. Based on it, we design a data structure suitable for dynamic graph algorithms. We employ the data structure to formulate new algorithms for several problems, including counting subgraphs of four vertices, reco...