On Face Antimagic Labelings of Plane Graphs

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 \( h: E \to \{1,2,\ldots,q\} \) 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 \( g_h(F) = \{a,a+d,\ldots,a + (t – 1)d\} \). We deal with \( (a,d) \)-face antimagic labelings for a certain class of plane graphs.