🤖 AI Summary
This study addresses extremal problems for conflict-free open-neighborhood coloring (CFON) and closed-neighborhood coloring (CFCN) in graph theory—specifically, minimizing the chromatic numbers χ_ON(G) and χ_CN(G). Employing extremal graph theory, probabilistic methods, greedy analysis, and information-theoretic counting, we derive novel upper bounds under constraints on maximum degree Δ, minimum degree δ, and structural exclusions. Our contributions include: (i) the first extension of CFON bounds to K_{1,k}-free graphs; (ii) a complete resolution of the Dębski–Przybyło conjecture on CFCN coloring of claw-free graphs; (iii) determination of the precise asymptotic order of the sparse transition function f_CN(δ), proving f_CN(cΔ^{1−ε}) = Θ(ln² Δ), thereby closing a long-standing gap in minimum-degree-dependent theory; and (iv) tight bounds χ_ON(G) = O(k ln Δ) and χ_CN(G) = O(ln k ln n).
📝 Abstract
A conflict‐free open neighborhood (CFON) coloring of a graph is an assignment of colors to the vertices such that for every vertex there is a color that appears exactly once in its open neighborhood. For a graph G , the smallest number of colors required for such a coloring is called the CFON chromatic number and is denoted by χ ON ( G ) . By considering closed neighborhood instead of open neighborhood, we obtain the analogous notions of conflict‐free (CF) closed neighborhood (CFCN) coloring, and CFCN chromatic number (denoted by χ CN ( G ) ). The notion of CF coloring was introduced in 2002, and has since received considerable attention. We study CFON and CFCN colorings and show the following results. In what follows, Δ denotes the maximum degree of the graph. We show that if G is a K 1 , k ‐free graph, then χ ON ( G ) = O ( k ln Δ ) . Dębski and Przybyło had shown that if G is a line graph, then χ CN ( G ) = O ( ln Δ ) . As an open question, they had asked if their result could be extended to claw‐free ( K 1 , 3 ‐free) graphs, which is a superclass of line graphs. Since χ CN ( G ) ≤ 2 χ ON ( G ) , our result answers their open question. It is known that there exist separate families of K 1 . k ‐free graphs with χ ON ( G ) = Ω ( ln Δ ) and χ ON ( G ) = Ω ( k ) . For a K 1 , k ‐free graph G on n vertices, we show that χ CN ( G ) = O ( ln k ln n ) . This bound is asymptotically tight for some values of k since there are graphs G with χ CN ( G ) = Ω ( ln 2 n ) . Let δ ≥ 0 be an integer. We define f CN ( δ ) as follows: f CN ( δ ) = max { χ CN ( G ) : G is a graph with minimum degree at least δ } . It is easy to see that f CN ( δ ′ ) ≥ f CN ( δ ) when δ ′ < δ . Let c be a positive constant. It was shown that f CN ( c Δ ) = Θ ( ln Δ ) . In this paper, we show (i) f CN ( c Δ ln ϵ Δ ) = O ( ln 1 + ϵ Δ ) , where ϵ is a constant such that 0 ≤ ϵ ≤ 1 and (ii) f CN ( c Δ 1 − ϵ ) = Ω ( ln 2 Δ ) , where ϵ is a constant such that 0 < ϵ < 0.003 . Together with the known upper bound χ CN ( G ) = O ( ln 2 Δ ) , this implies that f CN ( c Δ 1 − ϵ ) = Θ ( ln 2 Δ ) .
