Equivalence Classes of Colorings: the Topological Viewpoint

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) .

Social Networks, 1994

In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.

2009

We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.

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.

2008

arXiv: Combinatorics, 2016

The Thue colouring of a graph is a colouring such that the sequence of vertex colours of any path of even and finite length in $G$ is non-repetitive. The change in the Thue number, $\pi(G)$, as edges are iteratively removed from a graph $G$ is studied. The notion of the $\tau$-index denoted, $\tau(G)$, of a graph $G$ is introduced as well. $\tau(G)$ serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph.

Graphs and Combinatorics, 2019

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we obtain structural characterization of connected {P 5 , K 4 , Kite, Bull}-free graphs which contains an induced C 5 and connected {P 6 , C 5 , K 1,3 }-free graphs that contains an induced C 6 . Also, we prove that {P 5 , K 4 , Kite, Bull}-free graphs that contains an induced C 5 and {P 6 , C 5 , P 5 , K 1,3 }free graphs which contains an induced C 6 are k-indicated colorable for all k ≥ χ(G). In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in .

An L(2, 1)-coloring of a simple connected graph G is an assignment f of non-negative integers to the vertices of G such that | f (u) − f (v)| ≥ 2 if d(u, v) = 1 and | f (u) − f (v)| ≥ 1 if d(u, v) = 2 for all u, v ∈ V(G). The maximum color assigned by f is called span of f. The span λ(G) of a graph G is the smallest k such that there exists an L(2, 1)coloring with span k. An L(2, 1)-coloring with span λ(G) is called a span coloring of G. If f is an L(2, 1)-coloring of G with span k, then h ∈ {0, 1, 2,. .. , k} is a hole if there is no vertex v in G such that f (v) = h. A no-hole coloring is an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is irreducible, if no color can be decreased and yield another L(2, 1)-coloring. A graph G is inh-colorable if there exists an irreducible no-hole coloring of G. An irreducible no-hole coloring is referred as inh-coloring. The inh-span of a graph G, denoted by λ inh (G) is the smallest number k such that there is an irreducible no-hole L(2, 1)-coloring of G with span k. Mycielskian µ(G) of a graph G with vertex set {v 1 , v 2 ,. .. , v n } is a graph obtained from G by adding n + 1 new vertices {u 1 , u 2 ,. . ., u n , w}, joining w to each u i , 1 ≤ i ≤ n, and joining u i to each neighbour of v i in G. In this paper, we determine λ(µ(P n)), λ(µ(C n)), and show λ inh (µ(P n)) = λ(µ(P n)), λ inh (µ(C n)) = λ(µ(C n)).

Discrete Applied Mathematics, 1994

An undirected graph G = (V; E) is a (Hasse{) diagram if there is a poset P = (V; <) and an orientationẼ of E such that (x; y) 2Ẽ i x < y in P and there is no z with x < z < y. We then write G = D P . A pair (x; y) 2Ẽ is called a covering and denoted by x y.