Hoffman colorability of graphs with smallest eigenvalue at least -2

Abstract: In accordance with the Cameron-Goethals-Seidel-Shult Classification Theorem, we extend the characterization of Hoffman colorability of line graphs from (Abiad, Bosma, Van Veluw, 2025) to all connected graphs with smallest eigenvalue at least $-2$; we give a characterization of Hoffman colorability of generalized line graphs, and we completely classify the Hoffman colorable exceptional graphs. The 245 Hoffman colorable exceptional graphs from this classification admit a natural partial ordering, and we determine the 29 graphs that are maximal in this respect, in a way similar to the classification of maximal ($E_8$-representable) exceptional graphs as described in (Cvetković, Rowlinson, Simić, 2004). Lastly, as a byproduct and also similarly as in (loc. cit.), we determine all 39 graphs that are maximal with respect to being representable in the $E_7$ root system.

• The paper classifies all connected non-trivially Hoffman-colorable graphs with smallest eigenvalue at least −2 as chromatically balanced generalized line graphs, one six-vertex sporadic graph, or 245 exceptional graphs organized under 29 maximal examples.
• For generalized line graphs, Hoffman colorability is equivalent to chromatic balance, meaning the parameters satisfy a_i = c − deg(v_i) for a common c at least the chromatic number of the line graph.
• The classification also identifies exactly ten connected non-trivially Hoffman-colorable graphs with fewer than three times as many vertices as chromatic number and makes their vector and quantum chromatic numbers computable through the Hoffman bound.

Context and motivation

The Hoffman bound $h(G) = 1 - \lambda_{\max}(G)/\lambda_{\min}(G)$ is a spectral lower bound on the chromatic number of a non-empty graph. A graph is Hoffman colorable when this bound is tight; graphs that are neither bipartite nor regular complete multipartite are called non-trivially Hoffman colorable. Building on the characterization of Hoffman colorable line graphs by Abiad, Bosma, and van Veluw (2025), De Bruyn and van Veluw extend the classification to all connected graphs with smallest eigenvalue at least $-2$, using the Cameron–Goethals–Seidel–Shult (CGSS) classification as the structural backbone: such graphs are either generalized line graphs or $E_8$-representable "exceptional" graphs (2603.03859).

The paper's central result is that connected non-trivially Hoffman colorable graphs with $\lambda_{\min} \ge -2$ are precisely:

• chromatically balanced generalized line graphs — generalized line graphs $L(G;a_1,\dots,a_n)$ where $a_i = c - \deg(v_i)$ for some $c \ge \chi(L(G))$;
• a single sporadic graph on six vertices (the line graph of a triangle with three pendant edges), which is moreover the unique connected non-trivially Hoffman colorable graph with smallest eigenvalue strictly greater than $-2$;
• 245 exceptional graphs, each a chromatic component of one of 29 maximal ones.

A useful consequence noted early in the paper: for Hoffman colorable graphs, the vector and quantum chromatic numbers coincide with the chromatic number by sandwiching ($h(G) \le \chi_v(G) \le \chi_q(G) \le \chi(G)$), so the quantum chromatic number becomes computable for this class.

Generalized line graphs

For generalized line graphs with $\lambda_{\min} = -2$, the equivalence between Hoffman colorability and being chromatically balanced is proved via an explicit Perron eigenvector argument: assigning weight 2 to vertices coming from edges of the root graph and 1 to cocktail party vertices yields $-2$0 with $-2$1, and equality holds exactly when all $-2$2 are equal. This cleanly subsumes the earlier line graph criterion, since line graphs of 1-factorable graphs are precisely the balanced case with all $-2$3.

For $-2$4, the authors prove that any such graph must be a line graph, and the only non-trivially Hoffman colorable example is the sporadic six-vertex graph. The proof uses a Galois-conjugacy lemma: if the minimal polynomial over $-2$5 of the largest eigenvalue of a component of a dichromatic component is even, then the whole graph is bipartite. Applied to Dynkin-type components $-2$6 arising when exactly one cocktail party pair exists, this forces bipartiteness.

Exceptional graphs

The exceptional case splits into four regimes, handled by different techniques.

Regular graphs and cones. Every regular non-trivially Hoffman colorable exceptional graph has ratio bound 3 or 4. There are exactly 17 with ratio bound 3 (chromatic components of $-2$7 or the Schläfli graph $-2$8) and 70 with ratio bound 4 (induced subgraphs of $-2$9 or the Chang graphs). The key tool is the Hoffman coclique graph $E_8$0, whose cocliques correspond to Hoffman colorings of induced subgraphs sharing the same ratio bound and smallest eigenvalue; maximal cocliques yield chromatically maximal graphs. Exceptional cone graphs (87 of them, 13 maximal) reduce to regular graphs via the cone characterization: $E_8$1 is Hoffman colorable iff $E_8$2 is regular, Hoffman colorable, with color classes of size $E_8$3. A second, independent treatment of the ratio-bound-3 layer uses the closure of the class $E_8$4 under deletion of 3-cocliques, producing a Hasse-type diagram whose downward paths encode Hoffman colorings; this diagram also reveals a complementation symmetry within the Schläfli graph.

Irregular graphs in $E_8$5. The 35 irregular non-trivially Hoffman colorable exceptional graphs switching-equivalent to line graphs of graphs on eight vertices are shown to be exactly the Seidel switches $E_8$6 where $E_8$7 is non-trivially Hoffman colorable and $E_8$8 is a Hoffman color class. These have color class sizes $E_8$9, and the four maximal examples are $\lambda_{\min} \ge -2$0, $\lambda_{\min} \ge -2$1, $\lambda_{\min} \ge -2$2, $\lambda_{\min} \ge -2$3. The correspondence relies on a coclique bound specific to $\lambda_{\min} \ge -2$4: every coclique has size at most 4, and equality forces 2-regularity — which also shows cone graphs cannot lie in $\lambda_{\min} \ge -2$5.

Remaining graphs. Graphs that are neither cones nor in $\lambda_{\min} \ge -2$6 must be induced subgraphs of one of the 43 maximal exceptional graphs of type (b) or (c), or of a maximal $\lambda_{\min} \ge -2$7-representable graph. As a byproduct, the authors determine that there are exactly 39 maximal $\lambda_{\min} \ge -2$8-representable graphs, computed via maximal cocliques in the $\lambda_{\min} \ge -2$9 compatibility graph; none of the 11 non-cone, non-$L(G;a_1,\dots,a_n)$0 members lies in $L(G;a_1,\dots,a_n)$1. Applying an algorithm based on the generalized coclique-intersection graph $L(G;a_1,\dots,a_n)$2 — whose adjacency condition accounts for components of $L(G;a_1,\dots,a_n)$3 failing to preserve $L(G;a_1,\dots,a_n)$4 — yields exactly 36 further non-trivially Hoffman colorable exceptional graphs, partitioned into four classes by their sets $L(G;a_1,\dots,a_n)$5 of Hoffman color class sizes: $L(G;a_1,\dots,a_n)$6 (17 graphs), $L(G;a_1,\dots,a_n)$7 (6, including $L(G;a_1,\dots,a_n)$8), $L(G;a_1,\dots,a_n)$9 (3, namely $a_i = c - \deg(v_i)$0), and $a_i = c - \deg(v_i)$1 (10). Notably, $a_i = c - \deg(v_i)$2 is the only maximal example admitting Hoffman colorings with two distinct color-class size profiles ($a_i = c - \deg(v_i)$3 versus $a_i = c - \deg(v_i)$4); the flexibility traces back to the disconnected dichromatic component $a_i = c - \deg(v_i)$5.

Application: graphs with fewer than $a_i = c - \deg(v_i)$6 vertices

Using the main theorem together with a lemma showing that any connected Hoffman colorable graph with $a_i = c - \deg(v_i)$7 has $a_i = c - \deg(v_i)$8 and at least two color classes of size 2, the authors classify all ten connected non-trivially Hoffman colorable graphs satisfying this sparsity condition: the sporadic six-vertex graph, $a_i = c - \deg(v_i)$9, $c \ge \chi(L(G))$0, and seven chromatic components of $c \ge \chi(L(G))$1 of orders 11 through 20. For the components with exactly one color class of size 5, the structure is encoded completely by a multiset $c \ge \chi(L(G))$2 of cardinality $c \ge \chi(L(G))$3 recording how pairs of small color classes attach to the distinguished vertex of the $c \ge \chi(L(G))$4 dichromatic component; nine admissible multisets arise up to symmetry.

Computational aspects and limitations

Several steps rely on computer enumeration in SageMath 10.6: maximal cocliques in the Hoffman coclique graphs of the Schläfli graph, Chang graphs, and $c \ge \chi(L(G))$5; maximal cocliques in the $c \ge \chi(L(G))$6 compatibility graph; and the coclique-deletion analysis of type (b)/(c) maximal exceptional graphs. Isomorphism testing against known families (e.g., detecting generalized line graphs via 31 forbidden induced subgraphs) filters the outputs. One step in the bijection between switches $c \ge \chi(L(G))$7 and orbits of Hoffman color classes was verified computationally rather than theoretically ($c \ge \chi(L(G))$8 for each relevant $c \ge \chi(L(G))$9). The classification of exceptional graphs with $-2$0 rests on exhaustive inspection of the 573 such graphs from Cvetković–Rowlinson–Simić, checking integrality of the Hoffman bound. These dependencies mean the classification is exact but partly certificate-based; no independent verification of the computations is provided within the paper.

Conclusion

The paper completes the program begun for line graphs by giving a full structural characterization of Hoffman colorability across the CGSS classes: a clean combinatorial condition (chromatic balance) for generalized line graphs, a finite explicit list of 245 exceptional graphs organized under 29 maximal elements, and a single sporadic exception at $-2$1. The derived classifications of the 29 maximal Hoffman colorable exceptional graphs and the 39 maximal $-2$2-representable graphs parallel the classical theory of maximal exceptional graphs. An open question left implicit is whether analogous characterizations exist beyond the $-2$3 regime, where no comparable structural classification is available.