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Auburn University

Academic institutionnorthamerica · us
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Research library212linked papers
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Selected work

Representative Papers

Balanced-chromatic number and Hadwiger-like conjectures

Aug 02, 2023arXiv.org

This work addresses the extension of Hadwiger’s conjecture to signed graphs. It introduces the *balanced chromatic number*—the minimum number of vertex subsets required such that no subset induces a negative cycle—as the central combinatorial tool. By establishing a quantitative relationship between the balanced chromatic number and the existence of a ( ilde{K}_t) subdivision, the authors prove an upper bound of (O(t^2)), specifically (frac{79}{2}t^2). They formulate and rigorously prove the *signed-graph analogue of Hadwiger’s conjecture*, demonstrating its equivalence to the classical conjecture. Furthermore, they generalize Kawarabayashi’s result on odd minors to the signed-graph setting and uncover a deep connection between the balanced chromatic number and the odd Hadwiger conjecture. The work unifies structural coloring, minor theory, and subdivision analysis for signed graphs, providing a novel framework for Hadwiger-type problems in signed graph theory.

6 citationsRead paper

Uniquely optimal codes of low complexity are symmetric

Aug 28, 2020arXiv.org

This study addresses the fundamental question of whether optimal codes in compact metric spaces necessarily exhibit symmetry, focusing on low-complexity uniquely optimal encodings. Method: Integrating tools from metric geometry, extremal combinatorics, and group action theory, we develop constructive coding design techniques, symmetry detection algorithms, and rigorous optimality proofs. Contribution/Results: We formulate and systematically verify the universal conjecture that every low-complexity uniquely optimal code admits a nontrivial symmetry. Through comprehensive case studies on canonical spaces—including the sphere and torus—we empirically and theoretically confirm that symmetry is a necessary condition for uniqueness and optimality under low complexity constraints. Our work establishes, for the first time, a deep structural connection between the geometry of the underlying metric space and the symmetry properties of its optimal codes. This yields novel principled guidance for efficient encoding design, advancing both theoretical understanding and practical construction of optimal codes in geometric settings.

3 citationsRead paper

Qalb: Largest State-of-the-Art Urdu Large Language Model for 230M Speakers with Systematic Continued Pre-training

Jan 13, 2026

Urdu presents significant challenges for existing large language models due to its complex morphology, Nastaliq script, and low-resource status. This work proposes a two-stage training strategy based on LLaMA-3.1 8B: first, continual pretraining on a 1.97-billion-word multilingual corpus comprising Urdu and English from diverse sources, followed by supervised fine-tuning on the Alif instruction dataset. This approach effectively mitigates catastrophic forgetting while enhancing language adaptation. By systematically integrating large-scale continual pretraining with instruction tuning—the first such effort for Urdu—the model achieves substantial improvements in both comprehension and generation capabilities. It establishes new state-of-the-art results across seven benchmark tasks, attaining a weighted average score of 90.34, which represents gains of 3.24 and 44.64 points over the previous best-performing model and the base LLaMA-3.1 8B-Instruct, respectively.

1 citationsRead paper
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