The fine triangle intersections for maximum kite packings

TL;DR

This paper investigates the intersection properties of maximum kite packings, establishing precise conditions for the possible overlaps in terms of blocks and triangles for various orders.

Contribution

It characterizes the fine triangle intersection problem for maximum kite packings across different congruence classes of v, providing exact descriptions of feasible intersection parameters.

Findings

Determines the exact intersection sets for v ≡ 0,1 mod 8.

Shows that for other congruence classes, all admissible intersections are possible.

Provides a complete characterization of the intersection problem for maximum kite packings.

Abstract

In this paper the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let $Fin(v)={(s,t):$ $\exists$ a pair of maximum kite packings of order $v$ intersecting in $s$ blocks and $s+t$ triangles$}$. Let $Adm(v)={(s,t): s+t\leq b_v, s,t$ are non-negative integers$}$, where $b_v=\lfloor v(v-1)/8\rfloor$. It is established that $Fin(v)= Adm(v)\setminus {(b_v-1,0),(b_v-1,1)}$ for any integer $v\equiv 0,1 ({\rm mod} 8)$ and $v\geq 8$; $Fin(v)=Adm(v)$ for any integer $v\equiv 2,3,4,5,6,7 ({\rm mod} 8)$ and $v\geq 4$.