Acyclic chromatic indices of planar graphs with large girth

Abstract

a b s t r a c tAn acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a ′ (G) of G is the smallest k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree ∆ has a ′ (G) = ∆ if there exists a pair (k, m) ∈ {(3, 11), (4, 8), (5, 7), (8, 6)} such that G satisfies ∆ ≥ k and g(G) ≥ m.

Acyclic Chromatic Indices of Planar Graphs with Girth At Least 4

“…The same result for outerplanar graphs was obtained by Hou et al [14]. Wang et al [17,19] proved that every planar graph G has a (G) = if there is a pair (k, g) ∈ { (3,11), (4,8), (5,7), (8,6), (12,5)} such that G satisfies ≥ k and g(G) ≥ g, and Hudák et al [12] proved that every planar graph G has a (G) = if g ≥ 12 or there is a pair (k, g) ∈ { (4,8), (5,7), (6,6), (10,5)} such that G satisfies ≥ k and g(G) ≥ g, independently. Dong and Xu [9] proved that if G is a planar graph with g(G) ≥ 4, then a (G) ≤ + 5.…”

Acyclic edge coloring of triangle-free 1-planar graphs

“…Fiamčik [10] and later Alon et al [2] made the following conjecture: Alon et al [1] proved that χ a (G) ≤ 64Δ(G) for any graph G. Molloy and Reed [15] improved this bound to that χ a (G) ≤ 16Δ(G). Něsetřil and Wormald [17] proved that χ a (G) ≤ Δ(G) + 1 for a random Δ(G)-regular graph G. The acyclic edge coloring of some special classes of graphs was also investigated, including subcubic graphs (Basavaraju and Chandran [4]; Fiamčik [10]; Skulrattanakulchai [21]), graphs with maximum degree 4 (Basavaraju and Chandran [3]), outerplanar graphs (Hou et al [13]; Muthu et al [16]), series-parallel graphs (Hou et al [12]; Wang and Shu [22]), and planar graphs (Cohen et al [8]; Dong and Xu [9]; Fiedorowicz et al [11]; Shu and Wang [19,20]; Wang et al [23]; Yu et al [25]). In particular, Conjecture 1.1 was confirmed for planar graphs without 3-cycles by Shu and Wang [19].…”

Acyclic chromatic index of IC-planar graphs

“…Alon et al [3] proved that there is a constant c such that χ a (G) ≤ ∆(G) + 2 for a graph G with girth at least c∆(G) log(∆(G)). Něsetřil and Wormald [22] proved that χ a (G) ≤ ∆(G) + 1 for a random ∆(G)-regular graph G. The acyclic edge coloring of some special classes of graphs has been studied widely, including graphs with maximum degree 4 (Basavaraju and Chandran [5]), subcubic graphs (Basavaraju and Chandran [4]; Fiamčik [11]; Skulrattanakulchai [25]), sparse graphs (Muthu et al [20]), series-parallel graphs (Hou et al [15]; Wang and Shu [28]), outerplanar graphs (Hou et al [16]; Muthu et al [19]), planar graphs (Cohen et al [8]; Dong and Xu [9]; Fiedorowicz et al [12]; Shu and Wang [23,24]; Wang et al [29]; Yu et al [30]).…”