Analogues of cliques for oriented coloring

Abstract

We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

Outerplanar and Planar Oriented Cliques

“…Motivated by this observation, Klostermeyer and MacGillivray [6] introduced the following notion. An absolute oriented clique, or simply an oclique, is an oriented graph…”

“…For (|C|, |C α x |, |C β y |) = (12,6,6), (11,6,6), (10,6,6), (10,6,5), (10,5,5), (9,5,5), (8,4,4), we have |S| ≤ 12 − |C|, using Lemma 4.9(b).…”

“…Now v 6 cannot be adjacent to v 3 as (G) = 3. To have weak directed distance from v 6 to v 2 and v 4 at most 2 we must have the edges v 4 v 6 and v 5 v 6 . This will create a K 2,3 -minor contradicting the outerplanarity of G. Hence, we are done with this case as well.…”

“…Hence, we must have |N + (v 1 )| = 3 and |N − (v 1 )| = 2. Now to have weak directed distance at most 2 between all the vertices of N + (v 1 ) and between all the vertices of N − (v 1 ), keeping the graph outerplanar, we must have the graph depicted in Figure 2j as a spanning subgraph of G. (x) For |G| = 7: It has been proved by Klostermeyer and MacGillivray [6] that G must contain the graph depicted in Figure 2k as a spanning subgraph.…”

“…For (|C|, |C α x |, |C β y |) = (8, 6, 6), (7,6,6), (7,6,5), (6,6,6), (6,6,5), (6,6,4), (6,5,5), we are forced to have…”