Linear arboricity conjecture for infinite graphs

The linear arboricity $\text{la}(G)$ 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 $\text{la}(G)\le \lceil (\mathrm{\Delta }(G)+1)\mathrm{/}2\rceil$ for every finite graph $G$, where $\mathrm{\Delta }(G)$ 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 $\text{latop}(G)$ 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 $k+1$.