An Improved Bound for the Linear Arboricity Conjecture

Abstract In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph G of maximum degree $$\Delta $$ Δ can be decomposed into at most "Equation missing" linear forests. (A forest is if all its components are paths.) 1988, Alon proved conjecture holds asymptotically. The current best bound due to Ferber, Fox Jain from 2020 who showed $$\frac{\Delta }{2}+ O(\Delta ^{0.661})$$ 2 + O ( 0.661 ) suffices for large enough . Here, we show admits a decomposition 3\sqrt{\Delta } \log ^4 \Delta 3 log 4 forests provided enough. Moreover, our result also in more general list setting, where edges have (possibly different) sets permissible Thus List was only recently shown hold asymptotically by Kim second author. Indeed, proof method ties together well-known Colouring Conjecture; consequently, error term matches known error-term Molloy Reed 2000. This follows as make two copies every colour then seek proper edge colouring avoid bicoloured cycles between copy; achieve this via clever modification nibble method.