相关论文: A conjecture on critical graphs and connections to…

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose…

In a recent work, Kaiser, Stehl\'ik and \v{S}krekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Ha and Van…

Let $I$ be a square-free monomial ideal in $R = k[x_1,\ldots,x_n]$, and consider the sets of associated primes ${\rm Ass}(I^s)$ for all integers $s \geq 1$. Although it is known that the sets of associated primes of powers of $I$ eventually…

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R.…

It was conjectured by Ohba and confirmed recently by Noel et al. that, for any graph $G$, if $|V(G)|\le 2\chi(G)+1$ then $\chi_l(G)=\chi(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We show that…

A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing…

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In…

The chromatic number $\chi$ of a graph is bounded from below by its clique number $\omega,$ but it can be arbitrary large. Perfect graphs are defined by $\chi=\omega$ for all induced subgraphs. An interesting relaxation are $\chi$-bounded…

Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has chromatic number at most $k+1$. This well known conjecture remained open for almost twenty…

We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(\omega)$ and thus has…

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $\chi_S(G)$. In this paper, $\chi_S$-critical graphs are introduced as the graphs $G$ such that…

Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $\chi(F)=r+1$ and there is an edge $e$ in $F$ such that $\chi(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least…

Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with…

In 1998, Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G) + \Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum…

Ohba has conjectured \cite{ohb} that if the graph $G$ has $2\chi(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More…

The celebrated Erd\H{o}s--Stone--Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1 - 1/(\chi(\mathcal{F})-1) + o(1)$, where $\chi(\mathcal{F})$ is the minimum chromatic number of a graph…

Let $h(G)$ denote the largest $t$ such that $G$ contains $K_t$ as a minor and $\chi(G)$ be the chromatic number of $G$ respectively. In 1943, Hadwiger conjectured that $h(G) \geq \chi(G)$ for any graph $G$. In this paper, we prove that…

Let $G=(V(G), E(G))$ be a multigraph with maximum degree $\Delta(G)$, chromatic index $\chi'(G)$ and total chromatic number $\chi''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $\chi''(G)\leq…

Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge…

This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for…