相关论文: Equivalence Classes of Colorings
Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic…
We explore a new type of replacement of patterns in permutations, suggested by James Propp, that does not preserve the length of permutations. In particular, we focus on replacements between 123 and a pattern of two integer elements. We…
In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two…
Let $L$ be an alternating prime non-split link in $S^3$. We use the category of flypes between reduced alternating diagrams for $L$ to classify involutions on $L$. As consequences, we show that the quotient of an alternating periodic link…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'ak and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is…
A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs…
We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…
The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these…
Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…
An equitable coloring of a graph is a proper coloring where the sizes of any two different color classes do not differ by more than one. A graph is IC-planar if it can be drawn in the plane so that no two crossed edges have a common…
A proper $s$-coloring of an $n$-vertex graph is \emph{equitable} if every color class has size $\lfloor{n/s}\rfloor$ or $\lceil{n/s}\rceil$. A necessary condition to have an equitable $s$-coloring is that every vertex $v$ appears in an…
We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several…
By means of color chord diagrams we establish a necessary and sufficient condition for $O$-topological equivalence of functions with one essentially critical point on oriented surfaces with edge. We also calculate the number of…
Arrangements of pseudolines are a widely studied generalization of line arrangements. They are defined as a finite family of infinite curves in the Euclidean plane, any two of which intersect at exactly one point. One can state various…
A proper $k$-coloring of vertices of an $n$-vertex graph is equitable if the size of every color class is $\lfloor n/k\rfloor$ or $\lceil n/k\rceil$. An extension of it to list coloring requires only that the size of every color class is at…
Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable.…
Kempe equivalence is a classical and important notion on vertex coloring in graph theory. In the present paper, we introduce several ideals associated with graphs and provide a method to determine whether two $k$-colorings are Kempe…
There are many variations on partition functions for graph homomorphisms or colorings. The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group…
Extending theorems of J. E. Greene [Invent. Math. 192 (2013), 717-750] and A. S. Lipson [Enseign. Math. (2) 36 (1990), 93-114], we prove that the equivalence class of a classical link L under mutation is determined by Goeritz matrices…
In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced…