Orientation‐based edge‐colorings and linear arboricity of multigraphs

The Goldberg–Seymour Conjecture for f $f$ -colorings states that the -chromatic index of a loopless multigraph is essentially determined by either weighted maximum degree or density parameter. We introduce an oriented version -colorings, where now each color class edge-coloring required to be orientable in such way every vertex v $v$ has indegree and outdegree at most some specified values g ( ) $g(v)$ h $h(v)$ . prove associated , $(g,h)$ -oriented chromatic satisfies formula. then present simple applications this result variations -colorings. In particular, we show Linear Arboricity holds k $k$ -degenerate multigraphs when least 4 − 2 $4k-2$ improving recent bound Chen, Hao, Yu graphs. Finally, demonstrate always equal its list coloring analogue.