Packing 3-Vertex Paths in 2-Connected Graphs

TL;DR

This paper constructs specific 2-connected, cubic, bipartite, planar graphs demonstrating limitations in packing disjoint 3-vertex paths, challenging assumptions about such packings in these graph classes.

Contribution

It introduces an infinite family of graphs with properties that restrict the maximum number of disjoint 3-vertex paths, highlighting new structural constraints.

Findings

Existence of graphs with fewer disjoint 3-vertex paths than vertices divided by three

Construction of infinite graph families with specific connectivity and planarity properties

Counterexamples to potential packing conjectures in certain graph classes

Abstract

We give a construction that provides infinitely many 2-connected, cubic, bipartite, and planar graphs G with 3k vertices and such that the number of disjoint copies of a 3-vertex path in G is less than k.