Dimension and Order Bounds for Isometric Embeddings of Graphs into Abelian Cayley Graphs, and the Abelian Dividend

Abstract: We investigate the minimum size of finite abelian Cayley graphs that admit an isometric embedding of a finite connected graph. While every connected graph on n vertices embeds isometrically into a binary Cayley graph of dimension at most n-1, the smallest possible abelian host has remained largely unexplored. We establish fundamental lower bounds showing that every binary host has dimension at least max(diam(G), floor(log2 n)), whereas every finite abelian host has order at least max(n, 2^diam(G)). Moreover, we prove that the minimum host order equals n if and only if G is itself an abelian Cayley graph. Exact binary dimensions are obtained for several important graph families. Hypercubes, complete graphs of order 2^k, and even cycles attain the lower bound. For stars we prove k_min(K1,q)=floor(log2 q)+1 using maximum sum-free sets, yielding an exponential improvement over the naive and isometric dimensions. For odd cycles we prove k_min(Cm)=m-1 for all m<17 and reduce the general case to a cyclic-interval lemma, showing that the universal upper bound is tight. Our computational contribution is a certified exhaustive census of all 995 connected graphs with 2<=n<=7 vertices under general abelian compactifications. The data reveal an "abelian dividend": 569 graphs (57 percent) admit a strictly smaller abelian host than the best binary host, 707 (71 percent) admit an optimal host containing a cyclic factor Zm with m>2, and only 17 graphs attain the theoretical order floor max(n,2^diam(G)). These results demonstrate that compact non-binary abelian hosts are typical rather than exceptional, while binary hosts remain the universal worst-case construction. 2020 MSC:05C12, 05C25, 05C30, 11B75, 20K01

• The paper establishes the lower bounds β(G) ≥ max(diam(G), ⌈log₂ n⌉) for binary dimension and ν(G) ≥ max(n, 2diam(G)) for abelian host order, with equality cases identified for important graph families.
• The paper shows that stars can have exponentially smaller binary dimension through sum-free sets, while odd cycles attain the universal dimension bound subject to a cyclic-interval conjecture.
• The paper’s certified census of 995 connected graphs with at most seven vertices finds smaller non-binary abelian hosts for 569 graphs, with cyclic factors appearing in 71% of the best hosts and compression reaching 9×.

Overview

This paper, by Fokam Souop and Bitjoka, studies the minimal size of finite abelian Cayley graphs that admit a given connected graph $G$ on $n$ vertices as an isometric subgraph. Building on a companion construction [(2607.07939)'s companion manuscript] that realizes every such graph isometrically in the binary host $\mathbb{Z}_2^{n-1}$, the paper asks how far below this universal bound one can go. Two extremal quantities are analyzed: the binary dimension $\beta(G)$, the least $k$ with $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$ isometrically, and the host order $\nu(G) = \min|\Gamma|$ over all abelian hosts. The paper contributes matching lower bounds, exact values for several families, and an exhaustive computational census of all 995 connected graphs on at most seven vertices that reveals a systematic advantage of non-binary hosts—the "abelian dividend."

Lower bounds

The dimension lower bound rests on a geodesic-independence lemma: if generators $s_1,\dots,s_d$ spell a geodesic word in $Cay(\mathbb{Z}_2^k,S)$, then every sub-multiset of the word is itself geodesic (a replacement argument using commutativity), so all $2^d$ subset sums are distinct—equivalently, the generators are linearly independent over $n$0. Since any pair of vertices at distance $n$1 yields such a geodesic word, and injectivity forces $n$2, the authors obtain

$n$3

Each term of the bound is tight: hypercubes $n$4 attain both terms simultaneously ($n$5), complete graphs satisfy $n$6 via the generating set $n$7, and even cycles $n$8 embed into $n$9 through their antipodal-pair cuts, recovering Djoković's classical embedding.

For general hosts, the currency is group order. The key observation is that any connected vertex-transitive graph on $\mathbb{Z}_2^{n-1}$0 vertices has diameter at most $\mathbb{Z}_2^{n-1}$1 (via 2-connectivity and Menger's theorem). Since a Cayley graph is vertex-transitive and must realize distance $\mathbb{Z}_2^{n-1}$2, injectivity plus this fact give

$\mathbb{Z}_2^{n-1}$3

The order floor admits a clean characterization: $\mathbb{Z}_2^{n-1}$4 if and only if $\mathbb{Z}_2^{n-1}$5 is itself a Cayley graph of an abelian group. The forward direction follows because an isometric bijection onto $\mathbb{Z}_2^{n-1}$6 maps edges to edges and vice versa; the converse is the identity embedding. This immediately resolves $\mathbb{Z}_2^{n-1}$7 for cycles ($\mathbb{Z}_2^{n-1}$8, so odd cycles collapse from binary host order $\mathbb{Z}_2^{n-1}$9 to $\beta(G)$0), complete graphs, circulants (including $\beta(G)$1 via $\beta(G)$2), and paths ($\beta(G)$3, attained by stretching into $\beta(G)$4).

Stars: an exponential gap via sum-free sets

A natural conjecture would be that graphs without even cycles require the naive dimension $\beta(G)$5. The star $\beta(G)$6 refutes this. Normalizing the center to $\beta(G)$7, the leaf labels form exactly the generating set $\beta(G)$8, and pairwise leaf distances of 2 force $\beta(G)$9 to be sum-free in $k$0; conversely, sum-freeness suffices for isometry. The maximum sum-free set in $k$1 has size $k$2 (the odd-weight vectors), giving the exact value

$k$3

In particular $k$4: even on trees, and even on partial cubes, the binary dimension can sit exponentially below both the naive and the isometric hypercube dimensions. A scoped corollary shows the naive method remains minimal within the $k$5-quotient family for odd-cycle-free graphs, but not in general.

Odd cycles: tightness of the universal bound

At the opposite extreme, odd cycles show the universal upper bound cannot be improved: $k$6 for all odd $k$7, with the general case reduced to a cyclic-interval lemma. The argument encodes linear dependencies among edge generators as a dependency code $k$8 containing $k$9. Isometry requires, for every cyclic arc $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$0, that $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$1. The interval lemma asserts that any nontrivial proper $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$2 violates this for some arc $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$3; it is proved when the support of $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$4 fits inside an arc of length at most $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$5, and verified exhaustively for all odd $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$6 (all $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$7 subsets at $G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$8). If rank$G \hookrightarrow Cay(\mathbb{Z}_2^k, S)$9, some nontrivial $\nu(G) = \min|\Gamma|$0 exists, and the lemma produces a word shorter than the true graph distance—a contradiction. Hence rank $\nu(G) = \min|\Gamma|$1 and $\nu(G) = \min|\Gamma|$2.

The full statement for all odd $\nu(G) = \min|\Gamma|$3 therefore depends on Conjecture 1 (the interval lemma in the covering-arc regime); the paper states this dependence explicitly rather than claiming unconditional generality. Combined with the stars result, the window $\nu(G) = \min|\Gamma|$4 is filled at both ends, with position governed by the rank $\nu(G) = \min|\Gamma|$5 of the cycle–class matrix computed by the quotient framework.

The abelian dividend: a census of small graphs

The empirical centerpiece is an exhaustive census of all 995 connected graphs on $\nu(G) = \min|\Gamma|$6 vertices, run with a certified pipeline whose compactification stage searches general finite-index sublattices (enumerated in Hermite normal form, including non-diagonal folds) and certifies each candidate by checking all $\nu(G) = \min|\Gamma|$7 distances against a breadth-first computation of the host's metric. Three methodological caveats are stated plainly:

• Every reported abelian host is a certified isometric embedding, not a heuristic estimate.
• Both sides of the comparison are algorithmic upper bounds—"strictly smaller" means the abelian pipeline certifiably beat the binary pipeline, not that optima are claimed.
• The enumeration was restricted (diagonal folds only) beyond free rank two for hard instances, which can only under-count compact hosts; the dividend is thus understated.

The headline numbers are striking:

Quantity	Value
Graphs with certified abelian host strictly smaller than best binary host	569 / 995 (57%)
Graphs tying (best host found a power of two)	426 / 995 (43%)
Best hosts containing a cyclic factor $\nu(G) = \min|\Gamma|$8, $\nu(G) = \min|\Gamma|$9	707 / 995 (71%)
Graphs attaining the order floor $s_1,\dots,s_d$0	17 / 995
Median compression among winners	1.6×
Maximum compression	9×

Two structural conclusions follow. First, compact non-binary hosts are the rule rather than the exception on small graphs, while the binary host retains its role as the universally guaranteed construction requiring no fold search. Second, the order floor characterizes highly structured hosts (cycles, paths, circulants)—it is not typical behavior. The authors' structural explanation is that free directions surviving the quotient fold isometrically at moduli far below the power of two a purely binary refolding requires, so cyclic factors arise by default rather than from rare global symmetry. On benchmark families the dividend is large: $s_1,\dots,s_d$1 compresses from binary order 64 to $s_1,\dots,s_d$2, $s_1,\dots,s_d$3 from 32768 to $s_1,\dots,s_d$4, and $s_1,\dots,s_d$5 from 16 to 6.

A caveat on trend: the strict-dividend fraction declines mildly from 65% at $s_1,\dots,s_d$6 to 56% at $s_1,\dots,s_d$7, and the authors explicitly decline to extrapolate, flagging persistence at larger orders—and interaction with density and girth—as open empirical questions.

Limitations and open problems

Three questions remain open, each bearing directly on a main result. First, the cyclic-interval lemma is proved only in the covering-arc regime and verified computationally for $s_1,\dots,s_d$8; until it is settled, $s_1,\dots,s_d$9 for arbitrary odd $Cay(\mathbb{Z}_2^k,S)$0 rests on Conjecture 1. Second, the Petersen graph—the smallest vertex-transitive non-Cayley graph—forces $Cay(\mathbb{Z}_2^k,S)$1 by the equality characterization, while the Clebsch-graph embedding gives $Cay(\mathbb{Z}_2^k,S)$2; the exact value of $Cay(\mathbb{Z}_2^k,S)$3 is open. Third, the census covers only $Cay(\mathbb{Z}_2^k,S)$4, and its comparison methodology bounds rather than determines optima, so the prevalence of the dividend at scale is unresolved. Additionally, the conservative sublattice enumeration means reported host sizes may exceed true optima even where certification succeeds.

Conclusion

The paper establishes the lower bounds $Cay(\mathbb{Z}_2^k,S)$5 and $Cay(\mathbb{Z}_2^k,S)$6, characterizes equality at $Cay(\mathbb{Z}_2^k,S)$7 as precisely the abelian Cayley graphs, and fills the dimension window at both ends—stars via maximum sum-free sets, odd cycles via the cyclic-interval lemma. The census demonstrates that for a majority of small graphs, general abelian hosts strictly outperform the universally guaranteed binary construction, with cyclic factors present in 71% of optimal hosts found. The residual gaps—the interval lemma for all odd $Cay(\mathbb{Z}_2^k,S)$8, the Petersen value, and large-$Cay(\mathbb{Z}_2^k,S)$9 behavior of the dividend—define the immediate agenda for this line of work.