Related papers: The Generalized Kauffman-Harary Conjecture is True

We consider combinatorial maps with fixed combinatorial knot numbered with augmenting numeration called normalized knot. We show that knot's normalization doesn't affect combinatorial map what concerns its generality. Knot's normalization…

Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1)…

The odd Hadwiger's conjecture, made by Gerads and Seymour in early 1990s, is an analogue of the famous Hadwiger's conjecture. It says that every graph with no odd $K_t$-minor is $(t-1)$-colorable. This conjecture is known to be true for $t…

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been…

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$…

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi…

Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities.…

If a knot has the Alexander polynomial not equal to 1, then it is linear $n$-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot…

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of…

We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed…

An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing…

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes…

A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that…

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many…

Four-Color Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of Four-Color Theorem based on Kuratowski's Theorem using some induction argument and give a description of the most complicated…

Motivated by the Erd\H{o}s-Faber-Lov\'asz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and…

Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot $J$ by introducing additional crossings to a diagram of $J \# -\!J$ along its axis of symmetry. If…

In this article we present the following new fact for prime p=11. For knots 6_2 and 7_2, mincol_{11} 6_2 = 5 = mincol_{11} 7_2, along with the following feature. There is a pair of diagrams, one for 6_2 and the other one for 7_2, each of…