A map from E (G) to {1, 2, 3, ..., t} is called a t-linear coloring if (V (G), φ(α)) is a linear forest for 1 ≤ α ≤ t. The linear arboricity la (G) of a graph G defined by Harary [9] is the minimum number t for which G has a t-linear coloring. Let G be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove that if G contains no 4-cycles and intersecting triangles...

We present efficient algorithms for three coloring problems on subcubic graphs. (A subcubic graph has maximum degree at most three.) The first algorithm is for 4-edge coloring, or more generally, 4-list-edge coloring. Our algorithm runs in linear time, and appears to be simpler than previous ones. The second algorithm is the first randomized EREW PRAM algorithm for the same problem. It uses O(n...

In this paper, we show that for every graph of maximum average degree bounded away from d, any (d + 1)-coloring can be transformed into any other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any 8-coloring of a planar graph into any other 8-coloring with a polynomial number of recolori...

In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is NP-hard, even for planar graphs of maximum degree 4, and we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Further, we show that every planar graph has a subdivision with one vertex per edge that admits an acyclic 3-colori...

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...