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...

A radiocoloring of a graph G is an assignment to each node, one of the colors 0, 1, 2, . . . λ such that all pairs of adjacent nodes, get colors which differ by at least two and no pair of nodes at distance two, get the same color. The radiocoloring problem (RCP) consists of determining the minimum λ for a given graph. In this paper, we first implement the PARC (Parallel Algorithm for Radiocolo...

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and H is a proper vertex coloring V → {1, 2, . . .} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The algorithmic and combinatorial propert...

We strengthen the revised GCH theorem by showing, e.g., that for λ = cf(λ) > iω, for all but finitely many regular κ < iω, it holds that “λ is accessible on cofinality κ” in some weak sense (see below). As a corollary, λ = 2 = μ > iω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many κ ∈ Reg∩iω. We strengthen previous results on the black box and the m...

the robust coloring problem (rcp) is a generalization of the well-known graph coloring problem where we seek for a solution that remains valid when extra edges are added. the rcp is used in scheduling of events with possible last-minute changes and study frequency assignments of the electromagnetic spectrum. this problem has been proved as np-hard and in instances larger than 30 vertices, meta-...

The span λ(G) of a graph G is the smallest k for which G’s vertices can be L(2, 1)-colored, i.e., colored with integers in {0, 1, . . . , k} so that adjacent vertices’ colors differ by at least two, and colors of vertices at distance two differ. G is full-colorable if some such coloring uses all colors in {0, 1, . . . , λ(G)} and no others. We prove that all trees except stars are full-colorabl...

The set of all simple graphs on n vertices and m edges f (n, m, λ) The maximum number of coloring of a graph with n vertices and m edges in at most λ colors (n, m)-graph A graph of order n and size m i 1 The maximum number of colorings of graphs of given order and size Below is the plan of this chapter. 1 2 The maximum number of colorings of graphs of given order and size • Definitions and nota...

We derive rigorous upper and lower bounds for the ground state entropy of the q-state Potts antiferromagnet on the honeycomb and triangular lattices. These bounds are quite restrictive, especially for large q. ∗email: [email protected] ∗∗email: [email protected] Nonzero ground state disorder and associated entropy, S0 6= 0, is an important subject in statistical mechan...

For a century ago, one of the most famous problems in mathematics was to prove the FourColor Problem. During the period that the Four-Color Problem was unsolved, which spanned more than a century, many approaches were introduced with the hopes that they would lead to a solution of this famous problem. In 1913, Birkhoff 1 defined a function P M,λ that gives the number of proper λ-colorings of a ...

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...