On compact k-edge-colorings: A polynomial time reduction from linear to cyclic

A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k−1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring is cyclic compact if there are two positive integers av, bv in {0, 1, · · · , k − 1} for every vertex v such that the colors incident to v are exactly {av, (av + 1)mod k, · · · , bv}. The problem k − CCCP is to determine whether a given graph admits a cyclic compact k-edge coloring. We show that the k − LCCP with possibly imposed or forbidden colors on some edges is polynomially reducible to the k − CCCP when k ≥ 12, and to the 12− CCCP when k < 12.