New Results on Difference Distance Magic Labelings

TL;DR

The paper addresses the problem of identifying and constructing difference distance magic labelings for oriented graphs (DDMOGs). It develops an imbalance-based linear-algebra framework using the skew-adjacency matrix $S$ and weight vectors to relate vertex labeling to structural balance, and introduces vertex coalescence and a weighted-sum operation to build larger DDMOGs from smaller ones. Key contributions include proving the existence of connected DDMOGs for every $n\ge5$, balancing imbalances by adding a vertex, and introducing new graph-operations (including windmill-wheeled families and weighted sums) to obtain both connected and disconnected DDMOGs. The results advance the combinatorial and algebraic toolkit for DDM labeling, with open directions on distinguishing DDMOGs from non-DDMOGs and extending the framework to broader labeling schemes.

Abstract

A graph labeling assigns values to the components of a graph (vertices, edges, etc.). In particular, distance magic labelings have been widely studied in undirected graphs. In such a labeling, the vertices are labeled with unique values from one up to the number of vertices so that the sum of labels on the neighbors of any vertex is the same across all vertices. For oriented graphs, a related concept of distance difference magic has been studied. In a distance difference magic labeling, each vertex is given a unique value from one up to the number of vertices such that for each vertex the sums of the labels of vertices in the in-neighborhood minus the sums of the labels of vertices in the out-neighborhood equals zero. In this paper, we expand on this concept by showing a connected difference distance magic oriented graph on $n$ vertices exists for each integer $n \geq 5$. We also construct arbitrarily large difference distance magic oriented graphs from smaller ones using a new graph sum and exhibit a connection between linear algebra and this type of labeling.

Figures (13)

• Figure 1: Examples of various orientations on $\overset{\rightarrow}{W_4}$, with labelings and vertex weights included in (a) and (b)
• Figure 2: A digraph $D$ showing vertex imbalances. Note $imb(D) = 2$, while for instance $imb(v_3)=0$
• Figure 3: The weight and imbalance vectors of $\overset{\rightarrow}{W_4}$ along with its skew-adjacency matrix
• Figure 4: The process of reducing imbalance in Theorem \ref{['thm:imb1']}
• Figure 5: Illustrating the DDM labeling used in the proof of Theorem \ref{['thm:chain']}

Theorems & Definitions (28)

• Definition 2.1
• Definition 3.1
• Definition 3.2: Graph parameter
• Lemma 3.3
• proof
• Corollary 3.4
• proof
• Lemma 3.6
• proof
• Theorem 3.7