The Jumping Knight and Other (Super) Edge-Magic Constructions

Abstract

Let G be a graph of order p and size q with loops allowed. A bijective function f : V (G) ∪ E(G) → {i} p+q i=1 is an edge-magic labeling of G if the sum f (u) + f (uv) + f (v) = k is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that f (V (G)) = {i} p i=1 then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which is related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product in order to get edge-magic labelings of some families of graphs.

New Constructions for the n-Queens Problem

“…Note that when h is constant, D ⊗ h Γ is the Kronecker product. Many relations among labelings have been established using the ⊗ h -product and some particular families of graphs, namely S p and S k p (see for instance, [11,13,14,15,16,17,18]). The family S p contains all super edge-magic 1-regular labeled digraphs of order p where each vertex takes the name of the label that has been assigned to it.…”

Subdivided Stars Super (b, d) - Edge-Antimagic Total Graph Labelling

“…• [11], created the subdivided star subclasses T (n1, n2, n3, n4) and derived a super (b, 0)-edge-antimagic total labeling existence on them. In the similar paper, certain results related to w-trees subdivision are also verified.…”

The consecutively super edge-magic deficiency of graphs and related concepts

“…It is worth to mention that Acharya and Hegde [1] introduced the concept of strongly indexable graph that turns out to be equivalent to the concept of super edge-magic graph (see [8]). The study of super edge-magic labelings of graphs has proven to be crucial in the last two decades, since many relations with other types of labelings have been found (see [6]), and relations with other concepts such as Skolem and Langford sequences (see [9]), and dual shuffle primes and Jacobsthal sequences (see [11] and [14]).…”