The main result of this paper is an edge-coloured version of Tutte's $f$-factor theorem. We give a necessary and sufficient condition for an edge-coloured graph $G^c$ to have a properly coloured $f$-factor. We state and prove our result in terms of an auxiliary graph $G_f^c$ which has a 1-factor if and only if $G^c$ has a properly coloured $f$-factor; this is analogous to the "short proof" of the $f$-factor theorem given by Tutte in 1954. An alternative statement, analogous to the original $f$-factor theorem, is also given. We show that our theorem generalises the $f$-factor theorem; that is, the former implies the latter. We consider other properties of edge-coloured graphs, and show that similar results are unlikely for $f$-factors with rainbow components and distance-$d$-coloured $f$-factors, even when $d=2$ and the number of colours used is asymptotically minimal.

翻译：本文的主要结果是Tutte $f$-因子定理的边染色版本。我们给出了边染色图$G^c$存在适色$f$-因子的充要条件。我们通过辅助图$G_f^c$来陈述并证明该结果，该辅助图存在1-因子当且仅当$G^c$存在适色$f$-因子；这类似于Tutte在1954年给出的$f$-因子定理的"简短证明"。同时给出了另一个与原始$f$-因子定理类似的陈述。我们证明该定理推广了$f$-因子定理，即前者蕴含后者。我们还考虑了边染色图的其他性质，并表明对于具有彩虹分量的$f$-因子以及距离$d$染色$f$-因子，即使当$d=2$且使用颜色数渐近最小时，类似的结果也不太可能成立。