The conjunction of the linear arboricity conjecture and Lovász's path partition theorem

Orientation‐based edge‐colorings and linear arboricity of multigraphs

“…recently shown by Chen, Hao, and Yu [8] (originally announced in [7]) for simple graphs G. Then specializing to f t = , we will improve an old bound of Caro and Roditty [6] on the degree-t arboricity a G ( ) t of a simple graph G at least when G Δ( ) is large enough (Corollary 14), and our result is also more commonly applicable when G is a loopless multigraph. In Section 3.4, we will note results on the f -chromatic index that we can obtain using Theorem 2, including an approximation of the Goldberg-Seymour Conjecture 1 for f -colorings (Theorem 16).…”

“…Specializing to $$f=2$$, our result will give an upper bound on the linear arboricity of a loopless multigraph. In particular, in Section 3.3 we will show that the unsolved Linear Arboricity Conjecture holds for $$k$$‐degenerate loopless multigraphs $$G$$ when $${\rm{\Delta }}(G)\ge 4k-2$$ (Corollary 13), improving the bound $${\rm{\Delta }}(G)\ge 2{k}^{2}-2k$$ recently shown by Chen, Hao, and Yu [8] (originally announced in [7]) for simple graphs $$G$$. Then specializing to $$f=t$$, we will improve an old bound of Caro and Roditty [6] on the degree‐$$t$$ arboricity $${a}_{t}(G)$$ of a simple graph $$G$$ at least when $${\rm{\Delta }}(G)$$ is large enough (Corollary 14), and our result is also more commonly applicable when $$G$$ is a loopless multigraph.…”

“…Chen and Hao [7] recently proved the Linear Arboricity Conjecture 10 for k-degenerate simple graphs G when…”

“…They also proved that $$la(G)=\lceil {\rm{\Delta }}(G)\unicode{x02215}2\rceil $$ for all 2‐degenerate simple graphs $$G$$ when $${\rm{\Delta }}(G)\ge 5$$. Chen and Hao [7] recently proved the Linear Arboricity Conjecture 10 for $$k$$‐degenerate simple graphs $$G$$ when $${\rm{\Delta }}(G)\ge 2{k}^{2}-2k$$. (They also proved that $$la(G)=\lceil {\rm{\Delta }}(G)\unicode{x02215}2\rceil $$ when $${\rm{\Delta }}(G)\ge 2{k}^{2}-k$$, but our concern will be just showing that $$la(G)\le \lceil ({\rm{\Delta }}(G)+1)\unicode{x02215}2\rceil $$.)…”