Self-Reverse Distance Magic Labeling

• Self-reverse distance magic labeling is a graph labeling method that pairs each vertex label with its reverse through an automorphism, ensuring a constant neighbor sum.
• It reformulates the classic distance magic labeling into a zero-sum model, using quotient graphs and voltage assignments to analyze and compress graph symmetries.
• The theory supports explicit constructions in tetravalent graphs, yielding existence theorems and systematic classifications across various graph orders.

Searching arXiv for the specified paper to ground the article in the latest record. Self-reverse distance magic labeling is a refinement of distance magic labeling for regular graphs that incorporates an involutory symmetry pairing each label with its reverse. In the underlying distance magic setting, a graph of order $n$ admits a bijective labeling of its vertices by integers from $1$ to $n$ such that the sum of the labels on the neighbors of every vertex is constant. The self-reverse variant, introduced for regular graphs, requires that the reversal operation be realized by an automorphism of the graph, yielding a compact quotient description and a natural $\mathbb{Z}_2$-symmetry. The concept is developed in detail for tetravalent distance magic graphs, where it supports structural classification, existence theorems by order, explicit constructions, and computational enumeration (Kovář et al., 15 Jul 2025).

1. Distance magic labelings and the reverse operation

Let $G=(V,E)$ be a finite simple graph of order $n=|V|$. A bijection $f:V\to\{1,2,\dots,n\}$ is a distance magic labeling if there exists $k$ such that for every $v\in V$,

$w(v)=\sum_{u\in N(v)} f(u)=k.$

If $1$0 is $1$1-regular, the magic constant is uniquely determined by double counting:

$1$2

hence

$1$3

For such a labeling, the reverse labeling is defined by

$1$4

For every vertex $1$5,

$1$6

so $1$7 is again a distance magic labeling with the same magic constant (Kovář et al., 15 Jul 2025).

For regular graphs, an equivalent zero-sum model is often used. In this formulation, vertices are labeled with

$1$8

and the requirement is that the sum of the labels on the neighborhood of every vertex be $1$9. The passage between the classical model and the zero-sum model is given by the affine change

$n$0

This reformulation is central to the treatment of self-reversibility, because reversal becomes the sign change $n$1 (Kovář et al., 15 Jul 2025).

2. Definition of self-reverse distance magic labeling

Let $n$2 be a regular distance magic graph with labeling $n$3. For each vertex $n$4, let $n$5 be the unique vertex such that

$n$6

If $n$7 is odd, the unique vertex with label $n$8 is fixed; this is the central vertex. The partition $n$9 of $\mathbb{Z}_2$0 consists of the pairs $\mathbb{Z}_2$1 and, when $\mathbb{Z}_2$2 is odd, possibly one singleton (Kovář et al., 15 Jul 2025).

The labeling $\mathbb{Z}_2$3 is self-reverse if the involution $\mathbb{Z}_2$4 defined by $\mathbb{Z}_2$5 is an automorphism of $\mathbb{Z}_2$6. Equivalently, for every two distinct parts $\mathbb{Z}_2$7, the bipartite subgraph between $\mathbb{Z}_2$8 and $\mathbb{Z}_2$9 is either empty, a complete bipartite graph, or two disjoint edges. An equivalent formulation in the classical model is

$G=(V,E)$0

for all $G=(V,E)$1, where $G=(V,E)$2 is an involutory automorphism pairing each label with its reverse; when $G=(V,E)$3 is odd, $G=(V,E)$4 has one fixed point, namely the central vertex (Kovář et al., 15 Jul 2025).

A second equivalent criterion is phrased in terms of adjacency preservation under reversal: replacing $G=(V,E)$5 by its reverse $G=(V,E)$6 yields an equivalent labeling in the sense that adjacency between labels is preserved if and only if $G=(V,E)$7 is self-reverse. When $G=(V,E)$8 is self-reverse and $G=(V,E)$9 is even, the involution has only length-2 orbits. This symmetry-enhanced condition is what permits a compressed description of both the graph and the labeling.

3. Quotient graphs, $n=|V|$0-symmetry, and encoding of the labeling

Self-reverse labelings naturally define a $n=|V|$1-symmetry. Let $n=|V|$2 be the involution pairing each vertex with its reverse-labeled partner. The quotient graph $n=|V|$3 has one vertex for each orbit of $n=|V|$4: each pair $n=|V|$5 and, if $n=|V|$6 is odd, one singleton corresponding to the central vertex (Kovář et al., 15 Jul 2025).

The quotient encodes the interaction between the paired vertices in $n=|V|$7 by a three-way distinction:

Quotient feature	Configuration in $n=|V|$8	Label on quotient vertex
Solid edge	A perfect matching between two pairs, i.e. two disjoint edges	$n=|V|$9
Dashed edge	All cross edges present, i.e. a complete bipartite $f:V\to\{1,2,\dots,n\}$0	$f:V\to\{1,2,\dots,n\}$1
Dashed semiedge	$f:V\to\{1,2,\dots,n\}$2 adjacent to $f:V\to\{1,2,\dots,n\}$3	$f:V\to\{1,2,\dots,n\}$4 for the central orbit when present

In the language of graph covers, $f:V\to\{1,2,\dots,n\}$5 is a regular $f:V\to\{1,2,\dots,n\}$6-cover of this labeled, edge-colored quotient; solid edges have voltage $f:V\to\{1,2,\dots,n\}$7, dashed edges voltage $f:V\to\{1,2,\dots,n\}$8. The original labeling is recovered by assigning opposite signs within each fiber $f:V\to\{1,2,\dots,n\}$9, so one vertex receives $k$0 and the other $k$1, with solid and dashed edges determining whether neighbors in the lift have equal or opposite signs (Kovář et al., 15 Jul 2025).

A concrete example is given by the wreath graph $k$2 with vertices $k$3, $k$4, where each $k$5 is adjacent to $k$6. One non-degenerate self-reverse labeling assigns

$k$7

The corresponding quotient graph has labels $k$8 together with solid and dashed adjacencies and semiedges arranged so that lifting by $k$9 reproduces $v\in V$0 and recovers the labeling (Kovář et al., 15 Jul 2025).

4. Infinite families and general existence phenomena

Several known infinite families of tetravalent distance magic graphs admit self-reverse labelings. The wreath graphs $v\in V$1, $v\in V$2, have a natural self-reverse labeling

$v\in V$3

If $v\in V$4 is not divisible by $v\in V$5, every distance magic labeling of $v\in V$6 is self-reverse and degenerate, meaning that each pair $v\in V$7 has equal neighborhoods. If $v\in V$8 divides $v\in V$9, then $w(v)=\sum_{u\in N(v)} f(u)=k.$0 admits both degenerate and non-degenerate self-reverse labelings; for $w(v)=\sum_{u\in N(v)} f(u)=k.$1 there also exist non-self-reverse distance magic labelings (Kovář et al., 15 Jul 2025).

Cartesian products of cycles $w(v)=\sum_{u\in N(v)} f(u)=k.$2 that are distance magic have all distance magic labelings self-reverse. Tetravalent distance magic circulants $w(v)=\sum_{u\in N(v)} f(u)=k.$3 are known and admit self-reverse labelings; more generally, valency-4 distance magic circulants support self-reverse labelings. For direct products of two cycles that are distance magic, when one factor is divisible by $w(v)=\sum_{u\in N(v)} f(u)=k.$4, each connected component admits a self-reverse distance magic labeling (Kovář et al., 15 Jul 2025).

The principal existence theorem for connected tetravalent graphs is formulated by order. For $w(v)=\sum_{u\in N(v)} f(u)=k.$5, there exists a connected tetravalent graph of order $w(v)=\sum_{u\in N(v)} f(u)=k.$6 with a self-reverse distance magic labeling if and only if either $w(v)=\sum_{u\in N(v)} f(u)=k.$7 is even with $w(v)=\sum_{u\in N(v)} f(u)=k.$8, or $w(v)=\sum_{u\in N(v)} f(u)=k.$9 is odd with $1$00. The explicit exceptions are

$1$01

There exists such a graph with a non-degenerate self-reverse labeling if and only if either $1$02 or

$1$03

There exists a connected, non-wreath tetravalent graph of order $1$04 admitting a self-reverse labeling if and only if $1$05 with $1$06 (Kovář et al., 15 Jul 2025).

These results place self-reverse labelings not as an exceptional phenomenon but as a broad symmetry class within tetravalent distance magic theory. A plausible implication is that the quotient-based viewpoint is not merely descriptive; it materially enlarges the range of orders for which explicit constructions can be organized systematically.

5. The merge construction along cyclets

A general construction produces a new distance magic graph from two existing regular distance magic graphs. Let $1$07 and let $1$08 be $1$09-regular distance magic graphs with zero-sum labelings $1$10. Let

$1$11

be cyclets of even length $1$12. The merged graph

$1$13

is formed from the disjoint union of $1$14 and $1$15 by deleting the edges of $1$16 and $1$17 and adding cross edges $1$18 and $1$19 for all $1$20, with indices taken modulo $1$21. The resulting graph remains $1$22-regular (Kovář et al., 15 Jul 2025).

The label synthesis rule uses the partition

$1$23

together with three hypotheses: the bipartition $1$24 is balanced, the cyclet $1$25 is alternating with respect to $1$26, and the local sum match

$1$27

holds for all $1$28. Writing $1$29 and $1$30, one defines

$1$31

Then $1$32 is a distance magic labeling of the merged graph, with neighbor sums remaining $1$33 at each vertex (Kovář et al., 15 Jul 2025).

Self-reversibility is preserved under additional symmetry conditions. If $1$34 and $1$35 are self-reverse and $1$36 with either

$1$37

for all $1$38, or

$1$39

for all $1$40, then $1$41 is self-reverse on the merged graph. In quotient terms, this is gluing along an alternating cycle or path in each quotient in a manner that respects the $1$42-fibers, the solid/dashed structure, and the label differences (Kovář et al., 15 Jul 2025).

This construction is used to establish existence across orders. Odd orders at least $1$43 are obtained by starting from base examples of orders $1$44, $1$45, $1$46, and $1$47 and repeatedly merging with $1$48 along an appropriate $1$49-cycle in the quotient whose endpoints both carry semiedges and have a fixed label difference $1$50. Even orders are supplied in abundance by wreath graphs, with additional non-wreath constructions for $1$51, $1$52, $1$53, $1$54, and beyond (Kovář et al., 15 Jul 2025).

6. Structural constraints and classification up to order 30

The theory imposes strong structural constraints. In tetravalent graphs, if a self-reverse labeling $1$55 is degenerate, meaning that some vertex $1$56 is adjacent to both $1$57 and $1$58, then every $1$59 satisfies $1$60 and

$1$61

Consequently the graph has even order and is a wreath graph $1$62. For valency $1$63, balanced complete-bipartite interactions in every pair are precisely the degenerate case. Conversely, any self-reverse labeling on a connected tetravalent graph of odd order is automatically non-degenerate, because there is a single central vertex with label $1$64 and all other pairs are distinct and not twins (Kovář et al., 15 Jul 2025).

Computational enumeration determines all connected tetravalent graphs up to order $1$65 admitting non-degenerate self-reverse labelings, up to equivalence. For orders at most $1$66, the only such graphs are the wreath graphs $1$67 for $1$68, and non-degenerate self-reverse labelings occur only for $1$69 and $1$70. For $1$71, the paper reports the counts of nonequivalent non-degenerate self-reverse labelings, nonisomorphic graphs, and vertex-transitive examples (Kovář et al., 15 Jul 2025).

$1$72	#SR, #gr, #VT
16	48, 1, 1
17	0, 0, 0
18	136, 2, 1
19	0, 0, 0
20	66, 2, 1
21	57, 7, 0
22	0, 0, 0
23	675, 80, 0
24	11156, 9, 3
25	3063, 522, 0
26	31562, 37, 0
27	10951, 2647, 0
28	35402, 342, 0
29	68837, 22893, 0
30	229716, 4151, 1

Among the notable vertex-transitive examples are $1$73 of order $1$74, $1$75, $1$76, and the wreath graphs $1$77 and $1$78. Two additional vertex-transitive non-wreath examples are exhibited at orders $1$79 and $1$80 via their quotients; one of order $1$81 is a Cayley graph of $1$82, and the order-$1$83 example is vertex-transitive but not a Cayley graph, with a quotient resembling the Petersen graph with semiedges at each vertex (Kovář et al., 15 Jul 2025).

7. Significance, related directions, and open problems

Self-reverse labelings occupy a specific place within distance magic theory as an efficient symmetry class. They compress graph-label pairs via quotients and voltage assignments, thereby linking distance magic labeling to the theory of regular covers. They also identify precisely when reversing a labeling yields an equivalent labeling: this occurs exactly when the involution pairing labels with their reverses is an automorphism. In several major tetravalent families, including wreath graphs, circulants, and Cartesian and direct products of cycles, self-reversibility is pervasive; in some cases, such as the relevant Cartesian products of cycles, all distance magic labelings are self-reverse (Kovář et al., 15 Jul 2025).

The computational and structural results also isolate a set of unresolved problems. One is the odd vertex-transitive case: whether connected tetravalent vertex-transitive distance magic graphs of odd order exist, and if so whether any admit self-reverse labelings. A second is edge-transitive characterization: to characterize connected tetravalent edge-transitive distance magic graphs and determine which admit self-reverse labelings. A third is vertex-transitive characterization more generally, including the question of whether there exists a connected tetravalent vertex-transitive distance magic graph with no self-reverse labeling. A fourth concerns the Cayley versus non-Cayley distinction: whether infinitely many connected tetravalent vertex-transitive distance magic graphs that are not Cayley graphs exist, and whether infinitely many such non-Cayley examples admit self-reverse labelings (Kovář et al., 15 Jul 2025).

Further problems concern constructions and algorithmics. The merge construction may be extended to multiple disjoint cyclets and alternating paths, and one may develop algorithms to decide self-reversibility. The paper also points to an augmented single-graph construction that re-wires along two cyclets $1$84 inside one distance magic graph under the local sum compatibility

$1$85

preserving distance magic and forcing $1$86. This suggests that self-reverse distance magic labeling is not only a structural notion but also a constructive framework for generating and organizing broad classes of regular distance magic graphs (Kovář et al., 15 Jul 2025).