This note describes fast algorithms for computing the prime factors of connected, nonbipartite graphs with respect to the direct product, and of connected graphs with respect to the strong product. The complexities are O(mmin(n,∆)) for the direct product, and O(ma(G)∆) for the strong, where n is the order of the graph G to be factored, m its size, a(G) its arboricity, and ∆ its maximum degree. ...

We show that most algorithms from the literature on listing the triangles of a graph have a common abstraction. Our unifying framework highlights that these seemingly different algorithms are in fact instantiations of a single generic procedure, and even suggests some additional variants. More importantly, it yields parsimonious implementations that are in general more efficient than those desc...

A linear forest is a graph in which each connected component is a chordless path. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. When each path has length at most k a linear forest is a linear k-forest and lak(G) will denote the minimum number of linear k-forests partitioning E(G). We cl...

An easy count ing argument shows here that la(G)>f . f f t " d i f f icu l ty is in establishing the upper bound. This problem has received much attention; see Alon [1]. We show here how Alon's beautiful treatment for graphs with large girth allows us easily to handle random regular graphs. By the random regular graph G,,, (where rn is even) we mean a graph picked uniformly at random from the s...

We show that there exists a graph with nodes, where any forest with nodes is a node-induced subgraph of . Furthermore, the result implies existence of a graph with nodes that contains all -node graphs of fixed arboricity as node-induced subgraphs. We provide a lower bound of ! for the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted...

The paper generalizes the notion of locating-chromatic number of a graph such that it can be applied to disconnected graphs as well. In this sense, not all the graphs will have finite locating-chromatic numbers. We derive conditions under which a graph has a finite locating-chromatic number. In particular, we determine the locatingchromatic number of a uniform linear forest, namely a disjoint u...

A graph G is list point k-arborable if, whenever we are given a k-list assignment L(v) of colors for each vertex v ∈ V(G), we can choose a color c(v) ∈ L(v) for each vertex v so that each color class induces an acyclic subgraph of G, and is equitable list point k-arborable if G is list point k-arborable and each color appears on at most ⌈|V(G)|/k⌉ vertices of G. In this paper, we conjecture tha...