The incidence chromatic number of toroidal grids

Abstract

An incidence in a graph G is a pair (v, e) with v ∈ V (G) and e ∈ E(G), such that v and e are incident. Two incidences (v, e) and (w, f ) are adjacent if v = w, or e = f , or the edge vw equals e or f . The incidence chromatic number of G is the smallest k for which there exists a mapping from the set of incidences of G to a set of k colors that assigns distinct colors to adjacent incidences.In this paper, we prove that the incidence chromatic number of the toroidal grid T m,n = C m ✷C n equals 5 when m, n ≡ 0 (mod 5) and 6 otherwise.

Incidence coloring of Cartesian product graphs

“…The incidence coloring problem on the Cartesian product of some special classes of graphs has been extensively investigated, e.g., P m P n [7,8], P m C n [5,8], C m C n [10], P m K n,h [5], and Q n [9], where P m denotes a path of m vertices, C n a cycle of n vertices, K n,h a complete bipartite graph with two vertex sets of n and h, respectively, vertices and Q n the n-dimensional hypercube. In [11], Sun and Shiu showed that …”

On incidence coloring conjecture in Cartesian products of graphs

“…We also ask if the condition on a graph G (the first factor) can be somehow relaxed. As Sopena and Wu [16] showed, every toroidal grid T m,n , i.e. a Cartesian product of cycles C m and C n , admits an incidence coloring with at most ∆(T m,n ) + 2 colors.…”

“…These techniques are somehow limited as we preserve the structure of both initial colorings, whereas it may be more efficient to use all the available colors everywhere. In [16], the authors use pattern tiling, but their graphs are well structured in that case. Additional techniques of recoloring, but still not relying to the structure of the factors too much, would surely contribute to the field.…”

Genelleştirilmiş Fibonacci Çizgelerin Bazı Özellikleri

“…Çeşitli çizgelerin baglılık kromatik sayılarını hesaplanmak üzere çok sayıda çalışma yapılsa da, baglılık boyama, çizge kuramında halen yogun olarak çalışılan konulardan biridir (bkz. [11][12][13][14][15][16][17] ve referansları). Yukarıdaki tanımlar yardımıyla k. Böylece k < n/2 için I(F n (k)) baglılık çizgesi 2k + 1 renkle boyanabildiginden χ i (F n (k)) = 2k + 1 olur.…”