On Face Antimagic Labelings of Plane Graphs

Abstract

If $G=(V,E,F)$ is a finite connected plane graph on $|V|=p$ vertices, $|E|=q$ edges and $|F|=t$ faces, then $G$ is said to be $(a,d)$-face antimagic iff there exists a bijection $h:E\to \{1,2,\dots ,q\}$ and two positive integers $a$ and $d$ such that the induced mapping $g_h:F\to \mathbb{N}$, defined by $g_h(f)=\sum \{h(u,v):\text{edge }(u,v)\text{ surrounds the face }f\}$, is injective and has the image set $g_h(F)=\{a,a+d,\dots ,a+(t–1)d\}$. We deal with $(a,d)$-face antimagic labelings for a certain class of plane graphs.