Valuations of Plane Quartic Graphs

Abstract

We deal with $(a,d)$-face antimagic labelings of a certain class of plane quartic graphs. A connected plane graph $G=(V,E,F)$ is said to be $(a,d)$-$faceantimagic$ if there exist positive integers $a$ and $d$, and a bijection $g:E(G)\to \{1,2,\dots ,|E(G)|\}$ such that the induced mapping ${\phi }_g:F(G)\to N$, defined by ${\phi }_g(f)=\sum \{g(e):e\in E(G)\text{ adjacent to face }f\}$, is injective and ${\phi }_g(F)=\{a,a+d,\dots ,a+(|F(G)|–1)d\}$.