An asymptotically tight upper bound for the domination number of the $2$-token graph of path graphs

TL;DR

This work studies the domination number of the 2-token graph $F_2(P_n)$ of the path on $n$ vertices. It connects the known packing-number bound to a lower bound and provides an explicit, constructive upper bound via a translation-based coloring framework that maps to an auxiliary graph $I(n)$ isomorphic to $F_2(P_n)$. The main result asserts that for $n \ge 13$, the domination number satisfies $a(n-1) \le \gamma(F_2(P_n)) \le d(n)$, where $a(n)$ and $d(n)$ are piecewise quintic polynomials in $n$ depending on $n \,\bmod\,5$. The authors give a detailed, case-by-case construction for $D_r(h)$ achieving the upper bound and corroborate their bounds with exact computations up to $n=25$, observing that $\\gamma(F_2(P_n)) = d(n)$ in this range; they further note that the lower bound arises from $\\rho(F_2(P_n))$ and prior results on the packing number. The work advances the understanding of domination in token graphs and highlights an asymptotically tight bound of the form $\\gamma(F_2(P_n)) = \frac{n^2}{10} + O(n)$.

Abstract

In this note, we show that the domination number of the $2$-token graph of the path graph of order $n\geq 2$ is equal to $\frac{n^2}{10}+Θ(n)$.

Figures (6)

• Figure 1: $H(10)$ and $I(10)$; $I(10)$ is drawn with continuous segments and black vertices.
• Figure 2: The vertices of $A_0$ are enclosed in red squares, while those of $B_0$ are enclosed in green squares. We note that $N[D'_4(f)]\subseteq N[(D'_4(f)\setminus A_0)\cup B_0]$.
• Figure 3: The vertices of $A_1$ are enclosed in red squares, while those of $B_1$ are enclosed in green squares. We note that $N[D'_4(g)]\subseteq N[(D'_4(g)\setminus A_1)\cup B_1]$.
• Figure 4: The vertices of $A_2$ are enclosed in red squares, while those of $B_2$ are enclosed in green squares. We note that $N[D'_0(g)]\subseteq N[(D'_0(g)\setminus A_2)\cup B_2]$.
• Figure 5: The vertices of $A_3$ are enclosed in red squares, while those of $B_3$ are enclosed in green squares. We note that $N[D'_4(g)]\subseteq N[(D'_4(g)\setminus A_3)\cup B_3]$.

Theorems & Definitions (10)

• Theorem 2.1
• Conjecture 2.2
• Theorem 2.3
• Remark 2.4
• Proposition 3.1
• Corollary 3.2
• proof
• Proposition 3.3
• Proposition 3.4
• proof