Related papers: On the Quantum Chromatic Gap

We show that the combinatorial matter of graph coloring is, in fact, quantum in the sense of satisfying the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with…

The fastest known classical algorithm deciding the $k$-colorability of $n$-vertex graph requires running time $\Omega(2^n)$ for $k\ge 5$. In this work, we present an exponential-space quantum algorithm computing the chromatic number with…

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log…

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent…

Reed conjectured that the chromatic number of any graph is closer to its clique number than to its maximum degree plus one. We consider a recolouring version of this conjecture, with respect to Kempe changes. Namely, we investigate the…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

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…

A particular case of the Hindman--Galvin--Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums…

In 1975 Erd\H{o}s initiated the study of the following very natural question. What can be said about the chromatic number of unit distance graphs in $\mathbb{R}^2$ that have large girth? Over the years this question and its natural…

We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…

We prove that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$. That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded…

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…

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…

A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ which admit at least one such coloring. We disprove…

The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the chromatic number of the square $G^2$ of a planar graph $G$, depending on the maximum degree $\Delta(G)$ of $G$. The only case that the conjecture is resolved is…

We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The…

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…

A graph coloring has bounded clustering if each monochromatic component has bounded size. Equivalently, it is a partition of the vertices into induced subgraphs with bounded size components. This paper studies clustered colorings of graphs,…

The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones…

Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for…