Induced Embeddings of Graphs into Abelian Cayley Graphs

Abstract

For a finite graph $G$ on $n$ vertices, let $\eta(G)$ denote the least order of a finite abelian group $\Gamma$ for which $G$ is an induced subgraph of some Cayley graph of $\Gamma$. Babai and S\'os (1985) settled the worst-case order of magnitude: it is $\Theta(n^2)$. We treat $\eta$ instead as an invariant of the individual graph, minimised over all finite abelian groups rather than over the cyclic groups alone, which is the restriction implicit in the literature on representation numbers modulo $n$. We prove a local order floor: $\eta(G)$ is at least the maximum of $n$ and twice the largest independence number of a neighbourhood of $G$. This localises at an arbitrary vertex the correspondence of Babai and S\'os between induced stars and sum-free sets; a corollary of the classification of maximum sum-free sets in abelian groups does not lower this floor, but restricts which host orders are admissible and so prunes the search. We determine $\eta$ exactly for paths, where it equals $n+1$, and for complete bipartite graphs $K_{a,b}$, where it equals $2\max(a,b)$ and meets the floor. A Cartesian product bound gives $\eta(P_m \,\square\, P_m) = (1+o(1))n$. We report certified exact values of $\eta$ for $22$ graphs, computed over all abelian groups. Seventeen of the $22$ optimal hosts are cyclic, so on most of these graphs the cyclic restriction costs nothing; where it bites, however, it is expensive. A search restricted to cyclic groups returns $36$ for the Petersen graph against the true value $16$, and $59$ for the Frucht graph against $27$. The cost of the restriction is concentrated rather than diffuse, and we identify the graphs on which it is paid. We also determine $\eta$ exactly for the double stars $D_{q,q}$ with $2 \le q \le 6$, obtaining $5q$ in each case. Since $\eta(D_{6,6}) = 30$ exceeds $2n = 28$, no constant below $15/7$ can bound $\eta(T)/n$ over all trees.