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...

A c-coloring of the grid GN,M = [N] × [M] is a mapping into [c] such that no four corners forming rectangle have same color. In 2009 challenge was proposed to find 4-coloring G17,17. Though coloring produced, finding it proved be difficult. This raises question whether there some complexity lower bound. Consider following problem: given partial grid, can extended full c-coloring? We show this p...

Distributed Greedy Coloring is an interesting and intuitive variation of the standard Coloring problem. It still consists in coloring in a distributed setting each node of a given graph in such a way that two adjacent nodes do not get the same color, but it adds a further constraint. Given an order among the colors, a coloring is said to be greedy if there does not exist a node for which its as...

Introduced by Dvořák and Postle, the notion of DP-coloring generalizes list coloring helps to prove new results on coloring. We consider 1-defective 2-defective DP-colorings graphs with 2 colors. For j=1,2, we find exact lower bounds number edges in (j,2)-DP-critical (that is, that do not admit j-defective colors but whose every proper subgraph has such a coloring) given vertices. These also ho...

Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8. Disciplines Mathematics Comments Choi, Ilkyoo, Jan Ekstein, Přemysl Holub, and Bernard Lidický. "3-coloring triangle-free planar graphs with a precolored 9-cycle." 68 Europe...