ﻻ يوجد ملخص باللغة العربية
A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such coloring. The distinguishing threshold of $G$, denoted by $theta(G)$, is the minimum number $k$ such that every $k$-coloring of $G$ is distinguishing. In this paper, we study $theta(G)$, find its relation to the cycle structure of the automorphism group of $G$ and prove that $theta(G)=2$ if and only if $G$ is isomorphic to $K_2$ or $overline{K_2}$. Moreover, we study graphs that have the distinguishing threshold equal to 3 or more and prove that $theta(G)=D(G)$ if and only if $G$ is asymmetric, $K_n$ or $overline{K_n}$. Finally, we consider the graphs in the Johnson scheme for their distinguishing numbers and thresholds.
The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)to{1,2,dotsc,k}$ whose induced edge labels ${c(u),c(v)}$ are distinct for all edges $uv$. Previous work
Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, inclu
For graphs $G$ and $H$, let $G {displaystylesmash{begin{subarray}{c} hbox{$tinyrm rb$} longrightarrow hbox{$tinyrm p$} end{subarray}}}H$ denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$ in $G$. It is known that
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one
Consider the collection of hyperplanes in $mathbb{R}^n$ whose defining equations are given by ${x_i + x_j = 0mid 1leq i<jleq n}$. This arrangement is called the threshold arrangement since its regions are in bijection with labeled threshold graphs on