相关论文: Crossings, colorings, and cliques

Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we…

Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar…

Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound…

By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological…

In 2015, Brown and Erey conjectured that every $2$-connected graph $G$ on $n$ vertices with chromatic number $k\geq 4$ has at most $(x-1)_{k-1}\big((x-1)^{n-k+1}+(-1)^{n-k}\big)$ proper $x$-colorings for all $x\geq k$. Engbers, Erey, Fox,…

In an $r$-coloring of edges of the complete graph on $n$ vertices, how many edges are there in the largest monochromatic connected component? A construction of Gy\'arf\'as shows that for infinitely many values of $r$, there exist colorings…

A graph $G$ is {$k$-crossing-critical} if $cr(G)\ge k$, but $cr(G\setminus e)<k$ for each edge $e\in E(G)$, where $cr(G)$ is the crossing number of $G$. It is known that for any $k$-crossing-critical graph $G$, $cr(G)\le 2.5k+16$ holds, and…

Given graphs $H_1, H_2$, a {red, blue}-coloring of the edges of a graph $G$ is a critical coloring if $G$ has neither a red $H_1$ nor a blue $ H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G$ admits a critical coloring, but…

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It…

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random…

In the way of proving Kneser's conjecture, L\'{a}szl\'{o} Lov\'{a}sz settled out a new lower bound for the chromatic number. He showed that if neighborhood complex $\mathcal{N}(G)$ of a graph $G$ is topologically $k$-connected, then its…

A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq…

In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the…

The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…

In this paper, we prove a generalization of a conjecture of Erd\"{o}s, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph…

Given a graph $G$, let $\Delta_2(G)$ denote the maximum number of neighbors any two distinct vertices of $G$ have in common. Vu (2002) proposed that, provided $\Delta_2(G)$ is not too small as a proportion of the maximum degree $\Delta(G)$…

It is proved that for integers $b, r$ such that $3 \leq b < r \leq \binom{b+1}{2} - 1$, there exists a red/blue edge-colored graph such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed…

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…

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic number…