List colouring of graphs and generalized Dyck paths

TL;DR

This paper uncovers a connection between Catalan numbers and list colouring of complete graphs, providing explicit formulas for certain parameters related to graph colourability and paintability.

Contribution

It establishes a novel link between Catalan numbers and list colouring parameters of complete graphs, extending combinatorial understanding of graph colourability.

Findings

For complete graphs, the parameters m_c and m_p equal the number of certain dominated paths.

The parameters are explicitly computed using Catalan numbers for specific vertex mappings.

The work generalizes the combinatorial interpretation of Catalan numbers in graph theory.

Abstract

The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V(G) \to N$ is a mapping. For a nonnegative integer $m$, let $f^{(m)}$ be the extension of $f$ to the graph $ G \diamondplus \overline{K_m}$ for which $f^{(m)}(v)=|V(G)|$ for each vertex $v$ of $\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-paintable. We study the parameter $m_c(K_n, f)$ and $m_p(K_n,f)$ for arbitrary mappings $f$. For $\vec{x}=(x_1,x_2,\ldots,x_n)$, an $\vec{x}$-dominated path ending at $(a, b)$ is a monotonic path $P$ of the $a \times b$ grid from $(0,0)$ to $(a,b)$ such that each vertex $(i,j)$ on $P$ satisfies $i \le x_{j+1}$. Let $\psi(\vec{x})$ be the number of $\vec{x}$-dominated paths ending at $(x_n,n)$. By this definition, the Catalan number $C_n$ equals $ \psi((0,1, \ldots, n-1)) $. This paper proves that if $G=K_n$ has vertices $v_1, v_2, \ldots, v_n$ and $f(v_1) \le f(v_2) \le \ldots \le f(v_n)$, then $m_c(G,f)=m_p(G,f)=\psi(\vec{x}(f))$, where $\vec{x}(f)=(x_1, x_2, \ldots, x_n)$ and $x_i=f(v_i)-i$ for $i=1, 2,\ldots, n$. Therefore, if $f(v_i)=n$, then $m_c(K_n, f)=m_p(K_n, f)$ equals the Catalan number $C_n$.