Optimal Adjacent Vertex-Distinguishing Edge-Colorings of Circulant Graphs

Total, Equitable Total and Neighborhood sum distinguishing Total Colorings of Some Classes of Circulant Graphs

In this paper, we have obtained the total chromatic as well as equitable and neighborhood sum distinguishing total chromatic numbers of some classes of the circulant graphs.

图的联合顶点区分边着色的Vizing型定理

A Vizing-like theorem for union vertex-distinguishing edge coloring

本文研究了一种边着色变种，每条边被赋予颜色子集，顶点标签是其邻边颜色集的并集。研究了找到最小颜色数使每个顶点标签不同的问题，并证明对于任何图，该最小颜色数仅可能取3个值，针对路径、环和完全二叉树给出了精确值。

Union vertex-distinguishing edge colorings

The union vertex-distinguishing chromatic index $χ'_\cup(G)$ of a graph $G$ is the smallest natural number $k$ such that the edges of $G$ can be assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to the edges incident to each vertex is different. We prove that $χ'_\cup(G) \in \left\{ \left\lceil \log_2\left(n +1\right) \right\rceil, \left\lceil \log_2\left(n +1\right) \right\rceil+1 \right\}$ for a graph $G$ on $n$ vertices without a component of order at most two. This answers a question posed by Bousquet, Dailly, Duchêne, Kheddouci and Parreau, and independently by …

A Note On Vertex Distinguishing Edge colorings of Trees

A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by $χ\,'_s(G)$ called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds $χ\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or one …

Intersecting edge distinguishing colorings of hypergraphs

An edge labeling of a graph distinguishes neighbors by sets (multisets, resp.), if for any two adjacent vertices $u$ and $v$ the sets (multisets, resp.) of labels appearing on edges incident to $u$ and $v$ are different. In an analogous way we define total labelings distinguishing neighbors by sets or multisets: for each vertex, we consider labels on incident edges and the label of the vertex itself. In this paper we show that these problems, and also other problems of similar flavor, admit an elegant and natural generalization as a hypergraph coloring problem. An ieds-coloring (iedm-coloring,…

3-退化图的邻点可区分全染色

Adjacent vertex distinguishing total coloring of 3-degenerate graphs

本文验证了AVD全染色猜想对3-退化图成立，即每个3-退化图存在使用至多最大度加3种颜色的邻点可区分全染色。