On <span class="math inline">\(z\)</span>-cycle factorizations with two associate classes where <span class="math inline">\(z\)</span> is in <span class="math inline">\(\{4,4a\}\)</span> with even parameters

<p>Let <span class="math inline">\(K = K(a,p;\lambda_1,\lambda_2)\)</span> be the multigraph with: the number of vertices in each part equal to <span class="math inline">\(a\)</span>; the number of parts equal to <span class="math inline">\(p\)</span>; the number of edges joining any two vertices of the same part equal to <span class="math inline">\(\lambda_1\)</span>; and the number of edges joining any two vertices of different parts equal to <span class="math inline">\(\lambda_2\)</span>. The existence of <span class="math inline">\(C_4\)</span>-factorizations of <span class="math inline">\(K\)</span> has been settled when <span class="math inline">\(a\)</span> is even; when <span class="math inline">\(a \equiv 1 \ (\mbox{mod } 4)\)</span> with one exception; and for very few cases when <span class="math inline">\(a \equiv 3 \ (\mbox{mod } 4)\)</span>. The existence of <span class="math inline">\(C_z\)</span>-factorizations of <span class="math inline">\(K\)</span> has been settled when <span class="math inline">\(a \equiv 1 \ (\mbox{mod } z)\)</span> and <span class="math inline">\(\lambda_1\)</span> is even; when <span class="math inline">\(a \equiv 0 \ (\mbox{mod } z)\)</span>; and when <span class="math inline">\(z=2a\)</span> where both <span class="math inline">\(a\)</span> and <span class="math inline">\(\lambda_1\)</span> is even. In this paper, we give a construction for <span class="math inline">\(C_z\)</span>-factorizations of <span class="math inline">\(K\)</span> for <span class="math inline">\(z \in \{ 4,4a \}\)</span> when <span class="math inline">\(a\)</span> is even.</p>