Neighbor sum distinguishing total choosability of planar graphs without intersecting 4-cycles

Given a simple graph G, a proper total-k-coloring c:V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if ∑c(u)≠∑c(v) for any two adjacent vertices u,v∈V(G), where ∑c(v) denote the sum of the color of v and the colors of edges incident with v. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ′′(G). Pilśniak and Woźniak conjected that χΣ′′(G)≤Δ(G)+3 for any simple graph G. Let Lx(x∈V(G)∪E(G)) be a set of lists of real numbers and each of size k. The least number k for which for any specified collection of such lists, there exists a neighbor sum distinguish total coloring of G with colors from Lx for each x∈V(G)∪E(G) is called the neighbor sum distinguishing total choosability of G, denoted by chΣ′′(G). In this paper, it is proved that chΣ′′(G)≤max{Δ(G)+3,10} for any planar graph G without intersecting 4-cycles. © 2024