The problem of finding a maximum $2$-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum $t$-matching which excludes specified complete $t$-partite subgraphs, where $t$ is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete $t$-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most $2t-1$. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum $2$-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most $t+1$. Our algorithms are obtained from exploiting the discrete structure of restricted $t$-matchings and employing an algorithm for the Boolean edge-CSP.

翻译：寻找不含短圈的最大 $2$-匹配问题因其与哈密顿圈问题的关联而受到广泛关注。该问题被推广为寻找排除特定完全 $t$-部子图的最大 $t$-匹配，其中 $t$ 是固定正整数。这一推广问题的多项式可解性仍是开放问题。本文针对该问题的以下两种情况给出了多项式时间算法：第一种情况中，禁止的完全 $t$-部子图是边不交的；第二种情况中，输入图的最大度至多为 $2t-1$。第一种情况的结论扩展了Nam (1994) 的工作，该工作证明了寻找不含长度为四的圈（这些四圈是顶点不交的）的最大 $2$-匹配问题的多项式可解性。第二种结论则拓展了Bérczi和Végh (2010) 以及Kobayashi和Yin (2012) 的工作，后者主要关注最大度不超过 $t+1$ 的图。我们的算法通过利用受限制 $t$-匹配的离散结构，并采用布尔边CSP的算法而得到。