On the Maximum Number of Colors for Links

Advances in Applied Mathematics, 2008

In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformations on colored diagrams called Teneva transformations. Each of these transformations reduces the number of colors in the diagrams by one (up to a point). This allows us to further decrease the upper bounds on these minima. We conjecture on the value of these minima. We apply these transformations to rational knots.

Proceedings of the American Mathematical Society, 1989

We define a new polynomial invariant for a special class of dichromatic links. This polynomial generalizes the Jones polynomial.

Journal of Knot Theory and Its Ramifications, 2021

In this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

Discret. Math., 2021

A cyclic coloring of a plane graph $G$ is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph $G$ is its cyclic chromatic number $\chi_c(G)$. Let $\Delta^*(G)$ be the maximum face degree of a graph $G$. In this note we show that to prove the Cyclic Coloring Conjecture of Borodin from 1984, saying that every connected plane graph $G$ has $\chi_c(G) \leq \lfloor \frac{3}{2}\Delta^*(G)\rfloor$, it is enough to do it for subdivisions of simple $3$-connected plane graphs. We have discovered four new different upper bounds on $\chi_c(G)$ for graphs $G$ from this restricted family; three bounds of them are tight. As corollaries, we have shown that the conjecture holds for subdivisions of plane triangulations, simple $3$-connected plane quadrangulations, and simple $3$-connected plane pentagulations with an even maximum face degree, for regular subdivisions of simple $3$-connected ...

Two Fox m-colorings of a knot or link K are said to be equivalent if they differ only by a permutation of colors. The set of equivalence classes of m-colorings under this relation is the set Cm(K) of Fox mcoloring classes of K. We develop a combinatorical formula for |Cm(K)| for any knot or link K that depends only on the m-nullity of K. As a practical application, we determine the m-nullity, and therefore the value of |Cm(P (p,q,r) )|, for any any (p, q, r) pretzel link P (p,q,r) .

2005

In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a three-edge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the four-color conjecture at the end of the 19th century. We have also shown the applicability of our method to another well-known three edge-coloring conjecture on cubic graphs. Namely Tutte's conjecture that "every 2-connected cubic graph with no Petersen minor is 3-edge colorable". Hence the conclusion of this paper implies another non-computer proof of the four color theorem by using spiral-chains in different context.

Discrete Mathematics, 1999

A cyclic coloring is a vertex coloring such that vertices in a face receive di erent colors. Let be the maximum face degree of a graph. This article shows that plane graphs have cyclic 9 5 -colorings, improving results of Ore and Plummer, and of Borodin. The result is mainly a corollary of a best-possible upper bound on the minimum cyclic degree of a vertex of a plane graph in terms of its maximum face degree. The proof also yields results on the projective plane, as well as for d-diagonal colorings. Also, it is shown that plane graphs with = 5 have cyclic 8-colorings. This result and also the 9 5 result are not necessarily best possible.

Discrete Applied Mathematics, 2002

A mixed hypergraph is a pair H = (V; E ∪ A), where V is the vertex set and E (A) the edge (the co-edge) set of H. A legal colouring of H gives the same (di erent) colour(s) to at least two vertices of any co-edge (of any edge). The upper chromatic number of H is the maximum number (H) of colours that can be used in a legal colouring. After giving a general upper bound to (H), we here consider the co-hypergraph S = (X; ∅ ∪ T), where T is a (v3; b2)-conÿguration T over X . We prove that computing (S) is NP-hard, but there exists a polynomial-time algorithm returning a colouring with ¿ 5 6 (S) colours. We also provide an example showing that this approximation factor is tight. ?

2012

For any link and for any modulus $m$ we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).

2009

We develop a theorem for determining the p-colorability of any (m, n) torus knot. We also prove that any p-colorable (m, n) torus knot has exactly one p-coloring class. Finally, we show that every p-coloring of the braid projection of an (m, n) torus knot must use all of the p colors.