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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.31047 |
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| _version_ | 1866917547639570432 |
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| author | Ali, Annayat Raja, Rameez |
| author_facet | Ali, Annayat Raja, Rameez |
| contents | In this paper, we investigate the notion of the \textit{$λ$-chromatic polynomial} of a graph, which enumerates the number of distinct $L(2,1)$-colorings using colors from a prescribed finite set. We prove that the $λ$-chromatic polynomial of a graph with $n$ vertices is a monic polynomial of degree $n$ and provide a combinatorial interpretation via lattice point enumeration within the framework of inside-out polytopes. Moreover, we compute the $λ$-chromatic polynomial of complete graphs $K_n$ using lattice path enumeration, and we develop a block-gap technique to derive the $λ$-chromatic polynomials for complete bipartite and multipartite graphs. Our approach unifies geometric, combinatorial, and algebraic methods to provide a systematic treatment of $λ$-colorings across various families of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31047 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $λ$-Chromatic Polynomials and Polytope Geometry Ali, Annayat Raja, Rameez Combinatorics In this paper, we investigate the notion of the \textit{$λ$-chromatic polynomial} of a graph, which enumerates the number of distinct $L(2,1)$-colorings using colors from a prescribed finite set. We prove that the $λ$-chromatic polynomial of a graph with $n$ vertices is a monic polynomial of degree $n$ and provide a combinatorial interpretation via lattice point enumeration within the framework of inside-out polytopes. Moreover, we compute the $λ$-chromatic polynomial of complete graphs $K_n$ using lattice path enumeration, and we develop a block-gap technique to derive the $λ$-chromatic polynomials for complete bipartite and multipartite graphs. Our approach unifies geometric, combinatorial, and algebraic methods to provide a systematic treatment of $λ$-colorings across various families of graphs. |
| title | $λ$-Chromatic Polynomials and Polytope Geometry |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.31047 |