Related papers: Lower bounds for measurable chromatic numbers
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…
A coloring is called $s$-wide if no walk of length $2s-1$ connects vertices of the same color. A graph is $s$-widely colorable with $t$ colors if and only if it admits a homomorphism into a universal graph $W(s,t)$. Tardif observed that the…
We study the chromatic polynomials for m \times n square-lattice strips, of width 9 <= m <= 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed…
In this paper we investigate the chromatic number of the Grassmann graphs and of their powers, denoted $J_q(n,m,t)$. In this graph, the vertices correspond to the $m$-dimensional subspaces in $\mathbb{F}_q^n$ and two vertices are adjacent…
In this paper, we use various versions of Lov\'asz extension to systematically derive continuous formulations of problems from discrete mathematics. This will take place in the following context: (1) For combinatorial optimization problems…
We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $\Delta$ where each neighbourhood has density at most $d$, namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as…
The chromatic number of an planar graph is not greater than four and this is known by the famous four color theorem and is equal to two when the planar graph is bipartite. When the planar graph is even-triangulated or all cycles are greater…
The Minimum Sum Coloring Problem (MSCP) is derived from the Graph Coloring Problem (GCP) by associating a weight to each color. The aim of MSCP is to find a coloring solution of a graph such that the sum of color weights is minimum. MSCP…
One of the aims of this paper is to solve an open problem of Lovasz about relations between graph spectra and cut-distance. The paper starts with several inequalities between two versions of the cut-norm and the two largest singular values…
Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2{\omega} -…
In this paper we investigate a parameter of graphs, called the circular altitude, introduced by Peter Cameron. We show that the circular altitude provides a lower bound on the circular chromatic number, and hence on the chromatic number, of…
Using Ravenel's Thom spectrum $X(n)$, we introduce the concept of chromatic defect, which measures how far a spectrum is from being complex-orientable. We compute the chromatic defect of various examples of interest, such as finite spectra,…
Let $P$ be a set of $n$ points in general and convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this…
By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. The…
A measure theoretic approach of the problem that there exits a finite unit-distance graphs in the plane that are not five (or four) colorable.
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…
The chromatic threshold $\delta_\chi(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with…
An odd coloring of a graph $G$ is a proper vertex coloring $\varphi$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $\varphi^{-1}(c)\cap N(v)$ is odd. The concept of…
In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most $k$ if it has a fractional coloring in…
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very…