$\lambda$-Chromatic Polynomials and Polytope Geometry

In this paper, we investigate the notion of the $\lambda \mathit{\text{-chromatic}}\mathit{\text{polynomial}}$ of a graph, which enumerates the number of distinct $L(2,1)$-colorings using colors from a prescribed finite set. We prove that the $\lambda$-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 $\lambda$-chromatic polynomial of complete graphs $K_n$ using lattice path enumeration, and we develop a block-gap technique to derive the $\lambda$-chromatic polynomials for complete bipartite and multipartite graphs. Our approach unifies geometric, combinatorial, and algebraic methods to provide a systematic treatment of $\lambda$-colorings across various families of graphs.