On the k-edge magic graphs

TL;DR

This paper investigates the conditions under which maximal outerplanar graphs are k-edge magic, providing characterizations for specific cases and conjecturing a general property related to prime order graphs.

Contribution

It characterizes all maximal outerplanar graphs that are k-edge magic for certain parameters and proposes a conjecture for prime order graphs.

Findings

Maximal outerplanar graphs of order 4, 5, 7 are k-EM iff k ≡ 2 (mod p).

All maximal outerplanar graphs that are k-EM for k=3,4 are identified.

Characterization of (p, p-h)-graphs that are k-EM for h ≥ 0.

Abstract

Let $G$ be a graph with vertex set V and edge set E such that |V| = p and |E| = q. For integers k\geq 0, define an edge labeling f : E \rightarrow \{k,k+1,....,k+q-1\} and define the vertex sum for a vertex $v$ as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in which every vertex has a constant vertex sum (mod p), then G is said to be k-edge magic (k-EM). In this paper, we (i) show that all the maximal outerplanar graphs of order p = 4; 5; 7 are k-EM if and only if k\equiv 2 (mod p); (ii) obtain all the maximal outerplanar graphs that are k-EM for k = 3; 4; and (iii) characterize all (p; p-h)-graph that are k-EM for h\geq 0. We conjecture that all maximal outerplanar graphs of prime order p are k-EM if and only if k \equiv 2 (mod p).