Given any fixed non-negative integer values h and k, the L (h, k)-labelling problem consists
in an assignment of non-negative integers to the nodes of a graph such that adjacent nodes …

Given any fixed non-negative integer values h and k, the L (h, k)-labelling problem consists
in an assignment of non-negative integers to the nodes of a graph such that adjacent nodes …

After a brief historical account, a few simple structural theorems about plane graphs useful
for coloring are stated, and two simple applications of discharging are given. Afterwards, the …

We prove that for any planar graph G with maximum degree Δ, it holds that the chromatic
number of the square of G satisfies χ (G2)≤ 2Δ+ 25. We generalize this result to integer …

Wegner conjectured that the chromatic number of the square of any planar graph G with
maximum degree Δ⩾ 8 is bounded by χ (G2)⩽⌊ 32Δ⌋+ 1. We prove the bound χ (G2)⩽⌈ …

It is well known that every planar graph contains a vertex of degree at most 5. A theorem of
Kotzig states that every 3-connected planar graph contains an edge whose endvertices …

For a planar graph G, let Δ(G), g(G), and λ(G;p,q) denote, respectively, its maximum degree,
girth, and L(p,q)-labeling number. We prove that (1) λ(G;p,q)≤(2q-1)Δ(G)+4p+4q-4 if …

We provide a “how-to” guide to the use and application of the Discharging Method. Our aim
is not to exhaustively survey results proved by this technique, but rather to demystify the …

arXiv:2210.05915v2 [math.CO] 22 Apr 2023 Page 1 arXiv:2210.05915v2 [math.CO] 22 Apr
2023 Coloring, List Coloring, and Painting Squares of Graphs (and other related problems) …

Wang and Lih conjectured that for every g≥ 5, there exists a number M (g) such that the
square of a planar graph G of girth at least g and maximum degree Δ≥ M (g) is (Δ+ 1) …