Perfecting the Line Graph
Hartosh Singh Bal
Abstract
This paper introduces two canonical constructions that transform arbitrary finite graphs into perfect graphs: the symmetric lift $\mathrm{HL}'_2(G)$, which is purely structural and label-invariant, and the ordered lift $\mathrm{HL}_2(G)$, which depends explicitly on vertex labeling and encodes directional information. Both lifts arise as line graphs of bipartite double covers and are box-perfect. Equivalently, if $\BDC{G}$ denotes the canonical (Kronecker) bipartite double cover of $G$, then \[ \HL'_2(G) \;=\; L(\BDC{G}), \] the line graph of $\BDC{G}$. The symmetric lift $\mathrm{HL}'_2(G)$ forms a canonical 2-cover of the line graph $L(G)$. This involution decomposes $\mathrm{HL}'_2(G)$ into symmetric and antisymmetric components: the symmetric part recovers $L(G)$, while the antisymmetric part yields a signed graph $L^-(G)$, the antisymmetric line graph, with $+1/-1$ edges encoding consistent versus crossed overlaps. Thus, all adjacency and Laplacian eigenvalues of $L(G)$, with multiplicities, appear within those of $\mathrm{HL}'_2(G)$, despite $L(G)$ typically not being a subgraph. For regular graphs such as Paley graphs, this construction yields infinite families of sparse, highly structured regular and box-perfect graphs whose lifts contain large cliques (of size $d$ for $d$-regular bases). When the base graph is an expander, the symmetric lift preserves its spectral expansion while producing dense local clique structure. Similar behavior is observed computationally for random regular base graphs, suggesting a natural framework for the study of box-perfect random regular graphs. Finally, we generalize these constructions to parameterized lifts $\mathrm{HL}_{r,d}(G)$ and $\mathrm{HL}_{r,d}'(G)$ defined on ordered $r$-tuples connected by Hamming distance constraints, which structurally encode the base graph and remain box-perfect.