TY  - JOUR
ID  - Czap2024
LB  - Czap2024
AU  - Czap, Július
TI  - Facial graceful coloring of plane graphs
ST  - Facial graceful coloring of plane graphs
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2024
VL  - 44
IS  - 6
SP  - 815
EP  - 825
DO  - https://doi.org/10.7494/OpMath.2024.44.6.815
UR  - https://doi.org/10.7494/OpMath.2024.44.6.815
AB  - 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 \(c:V(G)\rightarrow\{1,2,\dots,k\}\) such that the induced edge coloring \(c^{\prime}:E(G)\rightarrow\{1,2,\dots,k-1\}\) defined by \(c^{\prime(uv)}=|c(u)-c(v)|\) is a facial edge coloring. The minimum integer \(k\) for which \(G\) has a facial graceful \(k\)-coloring is denoted by \(\chi_{fg}(G)\). In this paper we prove that \(\chi_{fg}(G)\leq 14\) for every plane graph \(G\) and \(\chi_{fg}(H)\leq 9\) for every outerplane graph \(H\). Moreover, we give exact bounds for cacti and trees.
KW  - facial edge coloring
KW  - plane graph
SN  - 1232-9274
ER  -