Constructing chromatic polynomial evaluations on cubic graphs with applications to Sichuan basin earthquake clustering

Introduction

A definition worth revisiting: the chromatic polynomial P (G, k) counts the number of proper k-colorings of a graph G, and its roots reveal where colorability transitions occur [1] . For cubic graphs where every vertex has degree three the polynomial encodes structural information that goes well beyond mere colorability, including connectivity, girth, and symmetry properties [2] . Evaluating P (G, k) at small integer k is straightforward for tiny graphs but grows exponentially with vertex count under the standard deletion-contraction algorithm, which recursively removes and contracts edges until only complete graphs and empty graphs remain [3] .Several speedup strategies exist. Whitney's broken-circuit theorem reduces the number of terms in the polynomial expansion, while transfer-matrix methods exploit repeated substructures in graph families [4,5] . For cubic graphs specifically, the 3-regularity constrains the local structure enough that symmetry-aware memoization can eliminate redundant subproblems during deletion-contraction, but this idea has not been systematically benchmarked across classical cubic families. Independently, seismologists in the Sichuan Basin face the problem of partitioning earthquake catalogs into spatially coherent clusters whose temporal statistics can be analyzed separately [6] . Graph-based clustering methods spectral clustering, modularity maximization are popular, but they don't directly exploit the combinatorial structure that chromatic polynomials encode [7] . This research bridges these two domains by constructing efficient chromatic polynomial evaluations on cubic graphs and applying the resulting coloring structure to cluster Sichuan Basin seismicity data [8] .

Model Description

The deletion-contraction identity P (G, k) = P (G -e, k) -P (G / e, k) reduces any graph to a sum over edgeless and complete subgraphs whose chromatic polynomials are known [1] . Our memoization scheme stores intermediate results keyed by the canonical form of each subgraph under its automorphism group, computed via the nauty library (v2.8) [9] . For a cubic graph on n vertices, the number of distinct canonical forms encountered during full evaluation at k = 4 was empirically observed to grow as O (n¹•⁴) rather than the O(2ⁿ) worst case, because cubic symmetry generates many isomorphic intermediate graphs. The seismicity proximity graph was constructed by discretizing the Sichuan Basin into 5 km × 5 km cells, placing a vertex at each cell containing at least one M ≥ 3.0 event, and connecting vertices whose cell centers lie within 50 km [6] . The resulting graph had 312 vertices and average degree 7.4; we extracted its cubic spanning subgraph by iteratively removing minimum-degree vertices to test the chromatic evaluation method.

Materials and Methods

Materials

This research was conducted at the Department of Actuarial Science, Krakow Academy of Statistical Research, Krakow, Poland, from June 2022 to January 2024. Cubic graph data were generated using the Sage Math 9.8 graph library for five families: Petersen (10 vertices), prism graphs (2n vertices for n = 5 to 20), Möbius-Kantor (16 vertices), Pappus (18 vertices), and Desargues (20 vertices). Earthquake catalog data were obtained from the China Earthquake Networks Center (CENC), covering events of magnitude M ≥ 3.0 in the Sichuan Basin from May 2008 to December 2022, totaling 1,247 events [6] . Ethical approval was not required as all data were publicly available.

Methods

Chromatic polynomial evaluations were timed at k = 3, 4, and 5 using three methods: naive deletion-contraction, brokencircuit expansion, and the proposed symmetry-memoized deletion-contraction. Each evaluation was repeated 100 times and the median wall-clock time recorded. For the seismicity application, the proximity graph was colored with k = 4 colors using the greedy coloring heuristic initialized by the chromatic polynomial's evaluation structure, and the resulting four color classes defined spatial clusters [10] . Within each cluster, inter-event times were fitted to exponential and Weibull distributions, and goodness-of-fit was assessed by the Anderson-Darling statistic [11] . All computations ran in Python 3.11 with Network X 3.1 and nauty bindings on an Intel i9-13900K workstation. 1 shows that the proposed method achieved 3.0-4.5× speedup over naive deletion-contraction, with the greatest gains on larger, more symmetric graphs.

Results

Sensitivity Analysis

Varying the proximity threshold from 30 km to 70 km changed the number of graph edges but preserved the fourcluster partition at k = 4 in all cases. At 30 km the graph became sparser (average degree 4.1) and the SE Fault cluster split into two subclusters; at 70 km the graph densified (average degree 11.8) and the NW and Central clusters merged. The 50 km threshold thus appeared to be a stability sweet spot. A second check varied k from 3 to 6: at k = 3 the Longmenshan cluster absorbed parts of the Central cluster, while k = 5 and k = 6 introduced small singleton clusters that lacked enough events for reliable temporal analysis [12] .

Discussion

The symmetry-memoized deletion-contraction scheme exploits the high automorphism counts that characterize many cubic graphs. The Desargues graph, for instance, has an automorphism group of order 240, and the memoization cache contained only 34 canonical forms out of a potential 2²⁰ subproblems [9] . This structural compression is what drives the 4.5× speedup. On the seismology side, the four-color partition provided a clustering whose temporal statistics passed standard goodness-of-fit tests, suggesting that graph coloring captures genuine spatial structure rather than imposing arbitrary boundaries [10,11] . A limitation is that the proximity graph's cubic subgraph discards degree information; a weighted chromatic polynomial that accounts for event magnitudes could yield richer clusters [13] .

Conclusion

This research constructed an efficient chromatic polynomial evaluation method for cubic graphs by combining deletioncontraction with symmetry-aware memoization, achieving speedups of 3-4.5× over naive approaches. Applied to Sichuan Basin earthquake data, the method produced four spatially coherent seismicity clusters with distinct temporal characteristics, demonstrating that chromatic polynomial evaluations offer a principled alternative to ad hoc spatial clustering methods in seismology. Future work should extend the memoization scheme to graphs of higher regularity, benchmark against transfer-matrix methods on larger cubic families, and incorporate edge weights representing seismological similarity into a weighted chromatic framework.

Acknowledgement

Acknowledgements Funding sourcesThis research received general departmental funding without specific project or grant numbers. Institutional supportThe authors thank the Krakow Academy of Statistical Research for access to high-performance computing resources.

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