The List Linear Arboricity of Digraphs
Yueping Shi, Ping Hu

TL;DR
This paper extends the bounds on the list linear arboricity of digraphs, showing that for large maximum degree, the minimum number of directed linear forests needed to decompose a digraph is close to its maximum degree, with a logarithmic error term.
Contribution
It generalizes the Lang and Postle bound for undirected graphs to directed graphs, providing a near-optimal upper bound in the list setting for large maximum degree.
Findings
Proves $la(D) \,\leq\, \Delta + 6\sqrt{\Delta} \log^4 \Delta$ for large $\Delta$.
Establishes the same bound in the list coloring setting.
Extends known bounds from undirected to directed graphs.
Abstract
A (directed) linear forest is a (di)graph whose components are (directed) paths. The linear arboricity of a (di)graph is the minimum number of (directed) linear forests required to decompose its edges. Akiyama, Exoo, and Harary (1980) proposed the Linear Arboricity Conjecture that for any graph of maximum degree . The current best known bound, due to Lang and Postle (2023), establishes for sufficiently large . And they proved this in the stronger list setting proposed by An and Wu. For a digraph , let its maximum degree be the maximum of all in-degrees and out-degrees of its vertices. Nakayama and P\'{e}roche (1987) conjectured that for every digraph . We extend Lang and Postle's result to…
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Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · Graph theory and applications
