相关论文: A note on fall colorings of Kneser graphs

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. In this paper, we answer some questions of \cite{dun} about some relations between fall…

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. We denote ${\rm Fall}(G)$ the set of all positive integers $k$ for which $G$ has a fall…

We say a proper coloring of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, thenit has a…

For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that every vertex has an equal number of vertices of each color in its closed neighborhood is called…

Given a proper coloring $f$ of a graph $G$, a b-vertex in $f$ is a vertex that is adjacent to every color class but its own. It is a b-coloring if every color class contains at least one b-vertex, and it is a fall-coloring if every vertex…

We show that any proper coloring of a Kneser graph $KG_{n,k}$ with $n-2k+2$ colors contains a trivial color (i.e., a color consisting of sets that all contain a fixed element), provided $n>(2+\varepsilon)k^2$, where $\varepsilon\to 0$ as…

A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz…

A totally silver coloring of a graph G is a k--coloring of G such that for every vertex v \in V(G), each color appears exactly once on N[v], the closed neighborhood of v. A totally silver graph is a graph which admits a totally silver…

Complete colorings have the property that any two color classes has at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers comes from complete colorings, with some additional requirement. In…

A k-role coloring of a graph G is an assignment of k colors to the vertices of G such that if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n…

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is…

A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a…

Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $\pi : V(H) \longrightarrow \{1,2,\ldots, k\}$ so that each color class induces a $K_2$-free subgraph. For a graph $G$ of order at least $2$, a $G$-free $k$-coloring of $H$…

A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a…

In this paper, we will introduce an special kind of graph homomorphisms namely semi-locally-surjective graph homomorphisms and show some relations between semi-locally-surjective graph homomorphisms and colorful colorings of graphs and then…

A (finite, undirected) graph is $(n,k)$-colourable if we can assign each vertex a $k$-subset of $\{1,2,\ldots,n\}$ so that adjacent vertices receive disjoint subsets. We consider the following problem: if a graph is $(n,k)$-colourable, then…

A Kneser representation KG(H) for a graph G is a bijective assignment of hyperedges of a hypergraph H to the vertices of G such that two vertices of G are adjacent if and only if the corresponding hyperedges are disjoint. In this paper, we…

A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours…

The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of an $n$ elements set, with two vertices adjacent if they are disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices…

The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…