Optimally edge-colouring outerplanar graphs is in NC

Abstract

We prove that every outerplanar graph can be optimally edge-coloured in polylogarithmic time using a polynomial number of processors on a parallel random access machine without write conflicts (P-RAM).

The Matching Book Embedding of Biconnected Outerplanar Graphs

“…The edge (v 1 v 2 ) is always unmarked and the edge (v n v 1 ) is marked if and only if n is even. The set of marked edges is a maximum matching, denoted by M(G) [16] . Fig.…”

“…Lemma 3.4. [16] Let ∆(G) be the maximum degree of the biconnected outerplanar graph G and ∆(G) > 3, ∆(G ′ ) be the maximum degree of the reduced graph G ′ . Then ∆(G ′ ) = ∆(G) − 1.…”

“…Note that the maximum degree of G decrease by 1 after deleting all edges of M(G), whereas the maximum degree of each block can decrease by more than 1 [16] , that is for any biconnected outerplanar graph G with ∆(G) = k (k ≥ 4), if the reduced graph G ′ is connected, then each block B i (1 ≤ i ≤ b) of the reduced graph G ′ is a graph with maximum degree at most k − 2.…”