Multithreshold multipartite graphs

Abstract

We call a graph G $G$ a k $k$‐threshold graph if there are k $k$ distinct real numbers θ 1 , θ 2 , … , θ k ${\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{k}$ and a mapping r : V ( G ) → double-struckR $r:V(G)\to {\mathbb{R}}$ such that for any two vertices u , v ∈ V ( G ) $u,v\in V(G)$, we have that u v ∈ E ( G ) $uv\in E(G)$ if and only if there are odd numbers θ i ${\theta }_{i}$ such that θ i ≤ r ( u ) + r ( v ) ${\theta }_{i}\le r(u)+r(v)$. The least integer k $k$ such that G $G$ is a k $k$‐threshold graph is called a threshold number of G $G$, and denoted by normalΘ ( G ) ${\rm{\Theta }}(G)$. The well‐known family of threshold graphs is a set of graphs G $G$ with normalΘ ( G ) ≤ 1 ${\rm{\Theta }}(G)\le 1$. Jamison and Sprague introduced the concept of k $k$‐threshold graph, and proved that normalΘ ( G ) ${\rm{\Theta }}(G)$ exists for every graph G $G$. They further obtained a number of interesting results on normalΘ ( G ) ${\rm{\Theta }}(G)$. In addition, they also proposed several unsolved problems and conjectures, including the following two. Problem: Determine the exact threshold numbers of the complete multipartite graphs. Conjecture: For all even n ≥ 2 $n\ge 2$, there is a graph G $G$ with normalΘ ( G ) = n ${\rm{\Theta }}(G)=n$ and normalΘ ( G c ) = n + 1 ${\rm{\Theta }}({G}^{c})=n+1$. This is equivalent to that for all odd n ≥ 3 $n\ge 3$, there is a graph G $G$ with normalΘ ( G ) = n ${\rm{\Theta }}(G)=n$ and normalΘ ( G c ) = n − 1 ${\rm{\Theta }}({G}^{c})=n-1$, where G c ${G}^{c}$ is the complement of G $G$.In this short paper, we give a partial solution of the problem and confirm the conjecture.

Threshold numbers of some complete multipartite graphs and their complements

“…Recently, Chen and Hao [2] gave a partial solution of Problem 1.2 which also confirmed Conjecture 1.1.…”

“…This conjecture was confirmed by Chen and Hao [2] (see Theorem 1.3). Note that In addition, Theorem 5.1 also improves the result of Puleo [17] providing an upper bound for Θ(K n×3 ).…”