Toughness and \((g,f)\)-factors in graphs with prescribed properties

<p>In this paper, we consider the relationship between toughness and the existence of <span class="math inline">\((g,f)\)</span>-factors with inclusion/exclusion properties. We obtain that if <span class="math inline">\(t(G) \geq \frac{(a+b)^{2}+2(b-a)-3}{4(a+1)}\)</span> with <span class="math inline">\(b > a \geq 2\)</span> and <span class="math inline">\(a \leq g(x) < f(x) \leq b\)</span> where <span class="math inline">\(a\)</span>, <span class="math inline">\(b\)</span> are two integers, then for any two given edges <span class="math inline">\(e_{1}\)</span> and <span class="math inline">\(e_{2}\)</span>, there exists a <span class="math inline">\((g,f)\)</span>-factor including <span class="math inline">\(e_{1}\)</span>, <span class="math inline">\(e_{2}\)</span>; and a <span class="math inline">\((g,f)\)</span>-factor including <span class="math inline">\(e_{1}\)</span> and excluding <span class="math inline">\(e_{2}\)</span>; as well as a <span class="math inline">\((g,f)\)</span>-factor excluding <span class="math inline">\(e_{1}\)</span>, <span class="math inline">\(e_{2}\)</span>.</p>