An odd c-coloring of a graph is proper such that each non-isolated vertex has color appearing an number times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski attracted considerable attention. Cranston investigated colorings graphs with bounded maximum average degree, conjectured every G mad(G)≤4c−4c+1 for c≥4, proved the conjecture c∈{5,6}. In partic...

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...

STATEMENT OF PROBLEM Speech bulbs and palatal lift prostheses are used to improve oral-nasal balance in speakers with hypernasality resulting from velopharyngeal dysfunction. Fabricating such speech prostheses is often a protracted process, and the nasopharyngeal impression can be uncomfortable for the client. PURPOSE The purpose of this study was to develop and test a modular palatal lift pr...

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...