Related papers: An eigenvalue bound for the fractional chromatic n…

The Erd\H{o}s-Faber-Lov\'{a}sz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability…

We introduce the signless 1-Laplacians and the dual Cheeger constants on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is…

Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic…

Given $0<\alpha\leq\pi$, ${\epsilon}>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$…

Our motivation is the question of how similar the $f$-colouring problem is to the classic edge-colouring problem, particularly with regard to graph parameters. In 2010, Zhang, Yu, and Liu gave a new description of the $f$-matching polytope…

We give a closed formula for Lovasz theta number of the powers of cycle graphs and of their complements, the circular complete graphs. As a consequence, we establish that the circular chromatic number of a circular perfect graph is…

Nordhaus and Gaddum proved, for any graph G, that the chromatic number of G plus the chromatic number of G complement is less than or equal to the number of vertices in G plus 1. Finck characterized the class of graphs that satisfy equality…

We study the Lovasz number theta along with two further SDP relaxations theta1, theta1/2 of the independence number and the corresponding relaxations of the chromatic number on random graphs G(n,p). We prove that these relaxations are…

Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that: \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G)…

We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional…

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the…

We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between half the number of vertices and the independence…

Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference…

To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint G\"ottingen 2006), we extend, in two directions, the Sobolev bound due to Arcang\'eli et al. (Numer Math 107, 181-211,…

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study…

In the setting of the integers, Granville, Harper and Soundararajan showed that the upper bound in Hal\'{a}sz's Theorem can be improved for smoothly supported functions. We derive the analogous result for Hal\'{a}sz's Theorem in…

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same…

We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-F\"uredi conjectured bound $\mathrm{q}(\mathcal{H})\le \Delta([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the…

Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have…

A theorem of Hakimi, Mitchem and Schmeichel from 1996 states that the edge arboricity arb(G) of a graph is bounded above by the acyclic chromatic number acy(G). We can improve this HMS inequality by 1, if acy(G) is even. We review also…