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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.
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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 has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.
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, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.
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, for every graph $H$, an asymptotic upper bound for the threshold function $p^{rm rb}_H=p^{rm rb}_H(n)$ of this property for the random graph $G(n,p)$ is $n^{-1/m^{(2)}(H)}$, where $m^{(2)}(H)$ denotes the so-called maximum $2$-density of $H$. Extending a result of Nenadov, Person, v{S}koric, and Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower bound for $p^{rm rb}_{K_k}$ for $kgeq 5$. Furthermore, we show that $p^{rm rb}_{K_4} = n^{-7/15}$.
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 color class. Given a graph $G$, let $lambda$ be an eigenvalue (of the adjacency matrix) of $G$ with multiplicity $k geq 1$. A vertex $v$ of $G$ is a downer, or neutral, or Parter depending whether the multiplicity of $lambda$ in $G-v$ is $k-1$, or $k$, or $k+1$, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi {em et al.}
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 $n$ vertices. Zaslavskys theorem implies that the number of regions of this arrangement is the sum of coefficients of the characteristic polynomial of the arrangement. In the present article we give a combinatorial meaning to these coefficients as the number of labeled threshold graphs with a certain property, thus answering a question posed by Stanley.
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