The simple graph threshold number $\sigma(r,s,a,t)$

Abstract

For $d \ge 1$, $s \ge 0$ a $(d, d+s)$-{\em graph} is a graph whose degrees all lie in the interval $\{d, d+1, \ldots, d + s\}$. For $r \ge 1$, $a \ge 0$, an $(r, r+a)$-{\em factor} of a graph $G$ is a spanning $(r, r+a)$-subgraph of $G$. An $(r, r+a)$-{\em factorization} of a graph $G$ is a decomposition of $G$ into edge-disjoint $(r, r+a)$-factors. A graph is $(r, r+a)$-{\em factorable} if it has an $(r, r+a)$-factorization. Let $\sigma(r, s, a, t)$ be the least integer such that, if $d \ge \sigma(r, s, a, t)$, then every $(d, d+s)$-simple graph $G$ is $(r,r+a)$-factorable with $x$ factors for at least $t$ different values of $x$. In this paper we evaluate $\sigma(r,s,a,t)$ for all values of $r, s, a$ and $t$. We also show that if $a \ge 2$ and $r \ge 1$, then, when $r$ is even and $a$ is odd, every $(d, d+s)$-simple graph $G$ has an $(r, r+a)$-factorization with $x$ factors if and only if $$ \frac{d+s}{r+a}\, < x \le \frac{d}{r}\,,$$ and we prove similar statements for other parities of $r$ and $a$.