We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G = (V, E) and positive node weights w : V → (0,∞), the goal is to find a proper node coloring of G whose color classes C1, C2, . . . , Ck minimize k ∑ i=1 maxv∈Ci w(v). We design a general framework which allows to convert approximation algorithms for standard node coloring into algorithms fo...

The dichromatic number of D, denoted by χ→(D), is the smallest integer k such that D admits an acyclic k-coloring. We use maderχ→(F) to denote if χ→(D)≥k, then contains a subdivision F. A digraph F called Mader-perfect for every subdigraph F′ F, maderχ→(F′)=|V(F′)|. extend octi digraphs larger class and prove it Mader-perfect, which generalizes result Gishboliner, Steiner Szabó [Dichromatic for...

An acyclic coloring of a graph is a proper coloring such that any two color classes induce a forest. A star coloring of a graph is an acyclic coloring with the further restriction that the forest induced by any two color classes is a disjoint collection of stars. We consider the behavior of these problems when restricted to certain classes of graphs. In particular, we give characterizations of ...

In cooperative game theory, a characterization of games with stable cores is known as one of the most notorious open problems. We study this problem for a special case of the minimum coloring games, introduced by Deng, Ibaraki & Nagamochi, which arises from a cost allocation problem when the players are involved in conflict. In this paper, we show that the minimum coloring game on a perfect gra...

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that appear in the closed neighborhood of u and v are distinct. Let χlid(G) be the minimum number of colors used in a locally identifying vertex-coloring of G. In ...

Given a bipartite graph G with n nodes, m edges and maximum degree ∆, we find an edge coloring for G using ∆ colors in time T +O(m log ∆), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k ≤ ∆. Together with best known bounds for T this implies an O(m log ∆ + m ∆ log m ∆ log ∆) edge-coloring algorithm which improves on the O(m log ∆+ m ∆ ...

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology [9], and the framework through which it was studied, we introduce the linear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be linearly colored in polynomial time by proposing a simple linear coloring ...

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

Motivated by the notion of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The colin...