Hoffman Colorings in Graph Spectral Theory

• Hoffman Colorings are optimal graph colorings attaining the spectral lower bound on chromatic number by achieving equality in Hoffman’s inequality.
• They exhibit structural rigidity through equitable or weight-regular partitions, which are key for both regular and irregular graphs.
• The theory extends to generalized line graphs, quantum and distance colorings, linking spectral extremality with combinatorial regularity.

Searching arXiv for papers on Hoffman colorings and related spectral coloring theory. arxiv_search(query="Hoffman colorings graph spectral chromatic number", max_results=10) arxiv_search({"query":"Hoffman colorings graph spectral chromatic number","max_results":10}) Hoffman colorings are optimal proper colorings that attain Hoffman’s spectral lower bound on chromatic number. For a non-empty graph $G$ with adjacency matrix $A$, largest adjacency eigenvalue $\lambda_{\max}(G)$, smallest adjacency eigenvalue $\lambda_{\min}(G)$, and chromatic number $\chi(G)$, the bound is

$$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$

A graph is called Hoffman colorable when equality holds, and any optimal coloring in that case is a Hoffman coloring (Abiad et al., 2024, Abiad et al., 26 Aug 2025, Bruyn et al., 4 Mar 2026). In contemporary graph theory the notion functions as an equality case for spectral lower bounds, a structural rigidity principle for regular and irregular graphs, and a bridge to equitable partitions, perfect colorings, Delsarte–Hoffman extremality, and several relaxed or generalized coloring parameters (Abiad et al., 2024, Potapov et al., 2024, Abiad et al., 15 Dec 2025, Guo et al., 2024).

1. Spectral definition and equality structure

The classical Hoffman bound is usually written in terms of the extreme adjacency eigenvalues, and recent work also treats $h(G)$ as the Hoffman number of $G$ (Abiad et al., 26 Aug 2025). A stronger refinement due to Hoffman is the $\kappa$-bound: if the adjacency eigenvalues are $\lambda_1\ge \cdots \ge \lambda_n$, then

$A$0

where $A$1 is the smallest integer such that

$A$2

This refinement has become important in later distance and quantum extensions (Abiad et al., 15 Dec 2025).

For regular graphs, equality in Hoffman’s bound is highly rigid. Each color class is a Hoffman coclique, all color classes have equal size, and every vertex has exactly $A$3 neighbors in every other color class (Abiad et al., 26 Aug 2025). In this regime a Hoffman coloring is therefore an equitable partition with particularly strong intersection-number constraints.

For connected irregular graphs, the regular equitable-partition picture is replaced by a weighted one. Using the positive Perron eigenvector for $A$4, the theory introduces weight-quotient matrices, weight-equitable partitions, and weight-regular partitions. If $A$5 is Hoffman colorable, then its color partition is weight-regular, every off-diagonal weight-intersection number equals $A$6, and the restriction of the positive eigenvector to each color class has the same norm (Abiad et al., 2024). This is the central extension from the classical regular case to general connected graphs.

2. Decomposition, composition, and structural classifications

A major structural result for general graphs is the Decomposition Theorem. If $A$7 is Hoffman colorable with color classes $A$8, and $A$9 is the induced subgraph on any subset of at least two color classes, then $\lambda_{\max}(G)$0 is again Hoffman colorable; moreover,

$\lambda_{\max}(G)$1

The restriction of the positive eigenvector of $\lambda_{\max}(G)$2 to $\lambda_{\max}(G)$3 remains a positive eigenvector of $\lambda_{\max}(G)$4 (Abiad et al., 2024). This induces strong local consequences: every vertex has at least one neighbor of every other color, every bipartite part induced by two colors has no isolated vertices, and a singleton color class must consist of a universal vertex.

The constructive counterpart is the Composition Theorem, which gives conditions for building a larger Hoffman colorable graph from a smaller one by adding one more color class while preserving the relevant spectral data. In the same work this leads to an algorithm for computing all connected Hoffman colorable graphs for prescribed numbers of vertices and colors (Abiad et al., 2024). Together, decomposition and composition turn Hoffman colorability into a recursive theory rather than a purely extremal equality condition.

These tools also yield complete classifications in several natural families. Cone graphs are completely characterized by a condition that the base graph be regular Hoffman colorable and satisfy the least-eigenvalue color-class constraint. Line graphs are also completely classified: if $\lambda_{\max}(G)$5 is connected with at least two edges, then $\lambda_{\max}(G)$6 is Hoffman colorable if and only if one of the following holds: $\lambda_{\max}(G)$7 is 1-factorable, $\lambda_{\max}(G)$8 for some $\lambda_{\max}(G)$9, $\lambda_{\min}(G)$0 is a path graph, $\lambda_{\min}(G)$1, or $\lambda_{\min}(G)$2 is the exceptional sporadic graph shown in the paper (Abiad et al., 2024).

3. Graphs with $\lambda_{\min}(G)$3, generalized line graphs, and strong regularity

The class of connected graphs with smallest eigenvalue at least $\lambda_{\min}(G)$4 is governed by the Cameron–Goethals–Seidel–Shult Classification Theorem: $\lambda_{\min}(G)$5 This dichotomy underlies a near-complete classification of Hoffman colorability in that spectral range (Bruyn et al., 4 Mar 2026).

For generalized line graphs, the characterization is chromatic balance. If $\lambda_{\min}(G)$6 is a connected generalized line graph with smallest eigenvalue exactly $\lambda_{\min}(G)$7, then

$\lambda_{\min}(G)$8

and the graph is Hoffman colorable if and only if it is chromatically balanced, meaning that all the quantities $\lambda_{\min}(G)$9 match the chromatic number (Bruyn et al., 4 Mar 2026). This extends the earlier line-graph classification.

The exceptional case is finite but nontrivial. There are exactly $\chi(G)$0 non-trivially Hoffman colorable exceptional graphs, and exactly $\chi(G)$1 of them are maximal with respect to the chromatic-component partial order. There are no non-trivially Hoffman colorable exceptional graphs with smallest eigenvalue strictly greater than $\chi(G)$2, and, as a byproduct, there are exactly $\chi(G)$3 graphs maximal with respect to being $\chi(G)$4-representable (Bruyn et al., 4 Mar 2026). The same classification implies that there are exactly ten connected non-trivially Hoffman colorable graphs with $\chi(G)$5.

In the strongly regular setting, Hoffman colorability has a geometric interpretation. For a primitive strongly regular graph, Hoffman colorability implies pseudo-geometricity, and if $\chi(G)$6 is Hoffman colorable then $\chi(G)$7 is also pseudo-geometric (Abiad et al., 26 Aug 2025). The same work proves that for every positive number $\chi(G)$8, there are only finitely many primitive strongly regular graphs with Hoffman number at most $\chi(G)$9. It also characterizes strong regularity among regular graphs by

$$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$0

with equality if and only if $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$1 is strongly regular (Abiad et al., 26 Aug 2025). Additional consequences are that a co-edge-regular Hoffman-colorable graph must be strongly regular, no strictly Neumaier graph can be Hoffman colorable, and every nontrivially Hoffman colorable connected 2-walk-regular graph is not uniquely vector colorable (Abiad et al., 26 Aug 2025).

4. Equitable partitions, perfect colorings, and Delsarte–Hoffman extremality

A closely related language is that of perfect colorings. A coloring is perfect if every vertex of a given color sees the same number of neighbors of each color; equivalently, its color classes form an equitable partition (Potapov et al., 2024, Potapov, 2022). In matrix form, if $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$2 is the adjacency matrix and $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$3 is the matrix of color-class indicators, then

$$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$4

for the quotient matrix $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$5, so the quotient eigenvalues belong to the spectrum of the graph (Potapov et al., 2024).

In a regular graph, the Delsarte–Hoffman bound for an independent set $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$6 states

$$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$7

where $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$8 is the valency, $$\chi(G)\ge h(G)\coloneqq 1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}.$$9, and $h(G)$0 is the minimal eigenvalue. Equality is equivalent to the indicator of $h(G)$1 being a perfect $h(G)$2-coloring; more generally, if every vertex of $h(G)$3 has at most $h(G)$4 neighbors inside $h(G)$5, then equality in

$h(G)$6

again forces a perfect $h(G)$7-coloring with an explicitly determined quotient matrix (Potapov et al., 2024). This is one of the clearest formulations of the idea that Hoffman-type extremality implies combinatorial regularity.

The same equality-only phenomenon persists for other spectral inequalities. For a regular connected graph, the Potapov generalization of the Hoffman bound for subsets with bounded average internal degree is attained if and only if the set is a perfect $h(G)$8-coloring. Equality in the Expander Mixing Lemma likewise holds if and only if the partition $h(G)$9 is equitable with the corresponding eigenvalue, and analogous statements are given for Cheeger-type cut bounds and for Boolean-function sensitivity bounds on Hamming graphs (Potapov, 2022). In this sense, Hoffman colorings sit inside a broader theory in which sharp spectral inequalities are attained exactly on equitable partitions.

This framework also connects to classical algebraic-combinatorial objects. Combinatorial designs, $G$0-designs, difference sets, Hadamard matrices, and bent functions can all be realized as perfect colorings of suitable graphs or multigraphs, especially Johnson graphs and Grassmann graphs (Potapov et al., 2024). The common mechanism is again the equitable-partition interpretation of extremal spectral structure.

5. Distance, quantum, and vector extensions

Hoffman’s spectral logic extends to distance colorings by working with the base graph $G$1 rather than the typically unrelated spectrum of $G$2. For a polynomial $G$3, the matrix $G$4 is supported on pairs of vertices at distance at most $G$5, which leads to a distance-$G$6 $G$7-bound. If $G$8, then the distance-$G$9 quantum chromatic number satisfies

$\kappa$0

and hence

$\kappa$1

The same paper rewrites the bound in a Hoffman-type ratio form and develops Binary Integer Linear Program and linear programming methods to optimize the polynomial choice (Abiad et al., 15 Dec 2025).

These distance bounds are sharp for several graph classes. The unified $\kappa$2-bound is sharp for $\kappa$3 in the distance-$\kappa$4 setting, for Lee graphs, for hypercubes, and for the Truncated Prism graph, where the distance-$\kappa$5 bound yields

$\kappa$6

while applying the $\kappa$7 bound to $\kappa$8 gives only $\kappa$9 (Abiad et al., 15 Dec 2025). The same work also extends the framework to the distance-$\lambda_1\ge \cdots \ge \lambda_n$0 vector chromatic number through polynomial bounds involving $\lambda_1\ge \cdots \ge \lambda_n$1, $\lambda_1\ge \cdots \ge \lambda_n$2, and either the diagonal maximum $\lambda_1\ge \cdots \ge \lambda_n$3 or the Perron-weighted diagonal average $\lambda_1\ge \cdots \ge \lambda_n$4.

A notable feature of this line of work is that it treats the $\lambda_1\ge \cdots \ge \lambda_n$5-bound, the quantum chromatic number, the vector chromatic number, and their distance variants in one polynomial framework. Conceptually, Hoffman-type colorings are thereby recast as part of a larger spectral theory in which equality and near-equality can be studied after polynomial filtering of the adjacency operator (Abiad et al., 15 Dec 2025).

6. Improper, hypergraph, and terminological extensions

The $\lambda_1\ge \cdots \ge \lambda_n$6-improper chromatic number $\lambda_1\ge \cdots \ge \lambda_n$7 admits a Hoffman-type lower bound due to Bilu: $\lambda_1\ge \cdots \ge \lambda_n$8 Recent work characterizes the equality case. If equality holds and $\lambda_1\ge \cdots \ge \lambda_n$9, then $A$00; if $A$01 is connected, each color class induces a $A$02-regular subgraph and the partition is weight-regular with respect to the Perron eigenvector; and if $A$03 is regular, the partition is equitable, with exactly $A$04 neighbors in its own color class and exactly $A$05 neighbors in each other color class (Guo et al., 2024). Thus the structural signatures of ordinary Hoffman colorings persist in the improper setting.

This improper theory is linked to ordinary Hoffman colorings through strong products. If $A$06 admits a Hoffman coloring, then

$A$07

admits a $A$08-improper Hoffman coloring, and the generalized Hoffman bound on the product coincides with the classical Hoffman bound on $A$09 (Guo et al., 2024). The same paper conjectures

$A$10

and proves the conjecture in special graph classes, including perfect graphs and graphs with chromatic number at most $A$11.

The spectral philosophy also extends beyond graphs. For hypergraphs, one can derive necessary spectral conditions for colorability by translating a hypergraph coloring into conflict bounds on auxiliary graphs, such as the underlying graph or subset graphs. In particular, if a $A$12-uniform hypergraph is $A$13-colorable, then

$A$14

where $A$15 is the average degree of the hypergraph and $A$16 is the smallest eigenvalue of the underlying graph; analogous necessary conditions are obtained for $A$17- and $A$18-uniform hypergraphs by combining several auxiliary graphs (Kenter, 2014). This is not a theory of Hoffman colorings in the strict graph-equality sense, but it is a direct extension of Hoffman’s spectral obstruction method.

A terminological distinction is also necessary. “Hoffman-London” graphs, introduced in the theory of graph homomorphisms, are target graphs $A$19 for which paths minimize $A$20 among trees of fixed order; the governing tool there is the automorphic similarity matrix and its increasing-columns property (Galvin et al., 29 Dec 2025). Likewise, ordered Szlam colorings are characterized by a dominance-by-translation structure and are described as conceptually analogous to “Hoffman coloring”-type ideas, but they are not themselves the spectral equality notion discussed above (Myzelev, 2024). In current usage, “Hoffman coloring” in the narrow sense refers specifically to a coloring attaining Hoffman’s eigenvalue lower bound on chromatic number.