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Facial graceful coloring of plane graphs

Czap Július

Opuscula Mathematica

2024

Vol. 44, no. 6

815--825

Let G be a plane graph. Two edges of G are facially adjacent if they are consecutive on the boundary walk of a face of G. A facial edge coloring of G is an edge coloring such that any two facially adjacent edges receive different colors. A facial graceful k-coloring of G is a proper vertex coloring c : V (G) → {1, 2, . . . , k} such that the induced edge coloring c′ : E(G) → {1, 2, . . . , k−1} defined by c′(uv) = |c(u)−c(v)| is a facial edge coloring. The minimum integer k for which G has a facial graceful k-coloring is denoted by χfg(G). In this paper we prove that χfg(G) ≤ 14 for every plane graph G and χfg(H) ≤ 9 for every outerplane graph H. Moreover, we give exact bounds for cacti and trees.