Oriented chromatic number of grids is greater than 7

The Oriented Chromatic Number of the Hexagonal Grid is 6

“…See [13] for a survey of the main results. Some bounds on the oriented chromatic number are known for several graph families like planar graphs [10], outerplanar graphs [11], graphs with bounded degree [11,8,15], Halin graphs [4], Cartesian products [12,14], hypergraphs [15], or grids [5,3,1].…”

“…Finding the exact value for the oriented chromatic number of the grid seems to be a surprisingly hard problem. Fertin, Raspaud, & Roychowdhury [5] first proved that 7 ≤ χ o (G 2 ) ≤ 11, the lower bound being subsequently improved to 8 by Dybizbański & Nenca [3]. The hexagonal grid H m,n is defined in [1] (only for m ≤ 2) as m rows of n hexagons (see Figure 2) and corresponds to the subgraph of G(m + 1, 2n + m) just containing the vertices v i,j that satisfy i − 1 ≤ j ≤ i + 2n, all edges of the kind {v i,j , v i,j+1 }, and edges of the kind {v i,j , v i+1,j } for even i + j (see Figure 1).…”

The oriented chromatic number of Halin graphs

“…The arc s 1 = (r(F 1 ), r(F 2 )) has colors (0, φ 1 (0)) = (0, 4) or (0, φ 2 (0)) = (0, 2) which are proper. The arc s 2 = (ll(F 1 ), f l(F 2 )) has colors (c 1 (ll(F 1 ), φ 1 (c 2 (f l(F 2 ))) = (3, φ 1 (1)) = (3, 5) or (3, φ 2 (1)) = (3,4) which are proper. Moreover, for every color x = 2, φ 1 (x) = φ 2 (x).…”

“…Oriented coloring has been studied in recent years [2,3,5,7,8,9,10,12,13,14,15], see [11] for a short survey of the main results. Several authors established or gave bounds on the oriented chromatic number for some families of graphs, such as: oriented planar graphs [9], outerplanar graphs [12,13], graphs with bounded degree three [8,12,14], k-trees [12], graphs with given excess [3], grids [4,5,15] or hexagonal grids [1].…”

Oriented chromatic number for Cartesian products Pm\Box Pn and Cm\Box Pn

“…Oriented coloring has been studied in recent years [1, 2, 6, 8-10, 12, 14, 16-20, 22], see [15] for a survey of the main results. Several authors established or bounded chromatic numbers for some families of graphs, such as oriented planar graphs [12,14], outerplanar graphs [12,17,18], graphs with bounded degree three [10,17,20], k-trees [17], Halin graphs [5,9], graphs with given excess [8] or grids [3,4,6,13,22].…”