AI 中文总结
本文推广线性森林猜想到无限图，证明有限与无限版本等价，并引入拓扑线性森林数，证明其与线性森林数相差至多1，且对围长至少2k的2k-正则图，拓扑线性森林数至多k+1。
AI 中文摘要
图 $G$ 的线性森林数 $\la(G)$ 是指将边集 $E(G)$ 分解为线性森林（即最大度至多 2 的森林）所需的最少森林数。线性森林猜想断言：对每个有限图 $G$，有 $\la(G)\leq\lceil(\Delta(G)+1)/2\rceil$，其中 $\Delta(G)$ 表示 $G$ 的最大度。我们将此猜想推广到最大度有限的无限图，并证明其有限版本与无限版本等价。我们通过要求线性森林不含拓扑圆，引入拓扑线性森林数 $\latop(G)$，并证明它与线性森林数至多相差 1。最后，我们证明每个围长至少为 $2k$ 的 $2k$-正则图，其拓扑线性森林数至多为 $k+1$。
英文摘要
The linear arboricity $\la(G)$ of a graph $G$ is the least cardinality of linear forests, that is, forests of maximum degree at most $2$, into which its edge set $E(G)$ can be decomposed. The Linear Arboricity Conjecture asserts that $\la(G)\leq\lceil(Δ(G)+1)/2\rceil$ for every finite graph $G$, where $Δ(G)$ denotes the maximum degree of $G$. We extend this conjecture to infinite graphs of finite maximum degree and prove that its finite and infinite versions are equivalent. We introduce topological linear arboricity $\latop (G)$ by requiring the linear forests to contain no topological circle, and show that it differs from linear arboricity by at most one. Finally, we prove that every $2k$-regular graph of girth at least $2k$ has topological linear arboricity at most $k+1$.
