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...

In this paper, we extend past work on Linear Scan register allocation, and propose two Extended Linear Scan (ELS) algorithms that retain the compiletime efficiency of past Linear Scan algorithms while delivering performance that can match or surpass that of Graph Coloring. Specifically, this paper makes the following contributions: – We highlight three fundamental theoretical limitations in usi...

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce study (i,j)-defective DP-colorings multigraphs. concentrate on sparse multigraphs consider fDP(i,j,n) — the minimum number edges that may have an n-vertex (i,j)-critical multigraph, is, multigraph G has no but whose every proper subgraph such coloring. F...

We give a linear-time algorithm to decide 3-colorability of triangle-free graph embedded in fixed surface, and quadratic-time output 3-coloring the affirmative case. The algorithms also allow prescribe coloring bounded number vertices.

The broadcast scheduling problem asks how a multihop network of broadcast transceivers operating on a shared medium may share the medium in such a way that communication over the entire network is possible. This can be naturally modeled as a graph coloring problem via distance-2 coloring (L(1, 1)-labeling, strict scheduling). This coloring is difficult to compute and may require a number of col...

The road coloring theorem states that every aperiodic directed graph with constant out-degree has a synchronized coloring. This theorem had been conjectured during many years as the road coloring problem before being settled by A. Trahtman. Trahtman’s proof leads to an algorithm that finds a synchronized labeling with a cubic worst-case time complexity. We show a variant of his construction wit...

Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a triangle-free gra...

A proper vertex coloring of a non oriented graph G = (V, E) is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V , then G...

An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u− v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in t...