On Neighbor-Distinguishing Index of Planar Graphs

Abstract

A proper edge coloring of a graph G without isolated edges is neighbor‐distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor‐distinguishing index of G is the minimum number ndi(G) of colors in a neighbor‐distinguishing edge coloring of G. Zhang, Liu, and Wang in 2002 conjectured that ndi (G)≤Δ(G)+2 if G is a connected graph of order at least 6. In this article, the conjecture is verified for planar graphs with maximum degree at least 12.

On the Neighbor Sum Distinguishing Index of Planar Graphs

“… for bipartite graphs and for graphs of maximum degree 3, while Greenhill and Ruciński proved it for almost all 4‐regular graphs (asymptotically almost surely), see . Recently it was also verified for planar graphs of maximum degree by means of the discharging method. Independently, Bonamy et al.…”

Neighbor sum distinguishing edge colorings of sparse graphs

“…Recently Hornák et al [10] confirmed this conjecture for planar graphs with maximum degree at least 12. Balister et al [2] proved Conjecture 1.1 for graphs with ∆(G) = 3 and for bipartite graphs.…”

Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

“…It was conjectured [27] that ′ a (G) ≤ Δ(G) + 2 for every connected graph G of order at least three different from the cycle C 5 . This was, for example, positively verified by Balister, Györi, Lehel, and Schelp [4] for bipartite graphs and for graphs of maximum degree 3, while Greenhill and Ruciński proved it asymptotically almost surely for random 4-regular graphs, see [10], and [12,13,25] for results concerning other particular graph classes. In general it is known that ′ a (G) ≤ 3Δ(G), [2], and ′ a (G) ≤ Δ(G) + O(log (G)), [4], for every graph G with no isolated edges.…”