[Paper Review] Magic labelings of distance at most 2

Rinovia Simanjuntak, Mona Elviyenti+3 more|arXiv (Cornell University)|Dec 30, 2013

Graph Labeling and Dimension Problems11 references3 citations

TL;DR

This paper investigates D-distance magic labelings for graphs, generalizing traditional distance magic labelings by considering arbitrary distance sets D ⊆ {0,1,2}. It establishes necessary and sufficient conditions for such labelings in cycles and graphs of diameter at most 2, proving non-existence for certain regular and multipartite graphs, and constructs infinite families of {1,2}-distance magic regular graphs with diameter 3.

ABSTRACT

For an arbitrary set of distances $D\subseteq \{0,1, \ldots, d\}$, a graph $G$ is said to be $D$-distance magic if there exists a bijection $f:V ightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) ={\sf k}$, where $N_D(x) = \{y \in V| d(x,y) \in D\}$. In this paper we study some necessary or sufficient conditions for the existence of $D$-distance magic graphs, some of which are generalization of conditions for the existence of $\{1\}$-distance magic graphs. More specifically, we study $D$-distance magic labelings for cycles and $D$-distance magic graphs for $D\subseteq\{0,1,2\}$.

Motivation & Objective

• To generalize distance magic labeling to arbitrary distance sets D ⊆ {0,1,2}, extending results from {1}-distance and {0,1}-distance magic labelings.
• To identify necessary and sufficient conditions for the existence of D-distance magic labelings in graphs, particularly for D = {2}, {0,2}, {1,2}, and {0,1,2}.
• To characterize which graphs—especially cycles, complete multipartite graphs, and regular graphs—admit D-distance magic labelings for various D.
• To explore open problems regarding the existence of such labelings in non-regular graphs and graphs of larger diameter.

Proposed method

• Define D-distance magic labeling as a bijection f: V → {1,2,…,v} such that the sum of labels over all vertices at distance d ∈ D from any vertex x is constant.
• Use neighborhood cardinality arguments: if |ND(x)| is odd and the graph is (D,r)-regular with odd r, then no D-distance magic labeling exists.
• Apply combinatorial constraints based on sum formulas: for a D-distance magic labeling to exist, the total sum of labels must satisfy modular conditions depending on |ND(x)| and the constant k.
• Construct explicit examples of {1,2}-distance magic labelings using symmetric graph constructions with vertices partitioned into sets connected via specific edge rules.
• Leverage known results on {1}-distance and {0,1}-distance magic labelings to derive conditions for {0,2}- and {1,2}-distance magic labelings.
• Use contradiction arguments (e.g., assuming two vertices have equal label sums but different labels) to prove non-existence for certain graph families.

Experimental results

Research questions

• RQ1 For which graphs does a D-distance magic labeling exist when D ⊆ {0,1,2}, particularly for D = {2}, {0,2}, {1,2}, and {0,1,2}?
• RQ2 What are the necessary and sufficient conditions for complete multipartite graphs to admit {2}-distance magic labeling?
• RQ3 Can infinite families of non-regular graphs or graphs of diameter ≥4 admit {1,2}-distance magic labeling?
• RQ4 Do graphs of diameter ≥3 admit {0,1,2}-distance magic labeling, or is diameter 2 necessary?
• RQ5 What characterizes trees that admit D-distance magic labeling for D ⊆ {0,1,2}?

Key findings

• Complete multipartite graphs are not {2}-distance magic because vertices in the same partite set have overlapping D-neighborhoods with odd size and odd regularity.
• The complete symmetric multipartite graph H_{n,p} is {0,2}-distance magic if and only if n is even or both n and p are odd.
• An odd-order r-regular graph of diameter 2 is {0,2}-distance magic if and only if r is even and 2 ≤ r ≤ n−2.
• The graph C₄[Ḡₙ] is {0,2}-distance magic for all n ≥ 1, showing that even-order graphs can admit {0,2}-distance magic labelings.
• An infinite family of n-regular graphs of order 2n+2 and diameter 3 exists that are {1,2}-distance magic, constructed via symmetric vertex and edge sets.
• Graphs of diameter 2 are not {d}-distance magic for d = 2, suggesting a general conjecture that graphs of diameter d are not {d}-distance magic.