On a list (k, l)-coloring of incidentors in multigraphs of even degree for some values of k and l

The problem of a list (k, l)-coloring of incidentors of a directed multigraph without loops is studied in the case where the lists of admissible colors for incidentors of each arc are integer intervals. According to a known conjecture, if the lengths of these interval are at least 2Δ + 2k - l - 1 for every arc, where Δ is the maximum degree of the multigraph, then there exists a list (k, l)-coloring of incidentors. We prove this conjecture for multigraphs of even maximum degree Δ with the following parameters: •l ≥ k + Δ/2; •l < k + Δ/2 and k or l is odd; • l < k + Δ/2 and k = 0 or l - k = 2.