Partitioning Harary graphs into connected subgraphs containing prescribed vertices

Olivier Baudon,Eric Sopena,Julien Bensmail

doi:10.46298/dmtcs.641

Abstract

A graph G is arbitrarily partitionable (AP for short) if for every partition (n_1, n_2, ..., n_p) of |V(G)| there exists a partition (V_1, V_2, ..., V_p) of V(G) such that each V_i induces a connected subgraph of G with order n_i. If, additionally, k of these subgraphs (k <= p) each contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k for short). Every AP+k graph on n vertices is (k+1)-connected, and thus has at least ceil(n(k+1)/2) edges. We show that there exist AP+k graphs on n vertices and ceil(n(k+1)/2) edges for every k >= 1 and n >= k.

Highlights

We denote by V (G) and E(G) the sets of vertices and edges, respectively, of a graph G
If X is a subset of V (G), G[X] denotes the subgraph of G induced by X
If G is a graph with a natural ordering of its vertices, for every vertex v of G, we denote by v+ the neighbour of v succeeding v in this ordering

Summary

Introduction

We denote by V (G) and E(G) the sets of vertices and edges, respectively, of a graph G. Given an integer k ≥ 1, the kth power of G, denoted by Gk, is the graph with the same vertex set as G, two vertices of Gk being adjacent if they are at distance at most k in G. The k-connected Harary graph on n vertices, denoted by Hk,n, has a vertex set {v0, v1, . If G is a graph with a natural ordering of its vertices (like powers of paths and cycles, or Harary graphs), for every vertex v of G, we denote by v+ According to Observation 4, it follows that the set of kth powers of cycles is a set of optimal AP+(2k − 1) graphs for every k ≥ 1

Partitioning powers of paths and cycles under prescriptions

Partitioning Harary graphs under prescriptions

Construction 2: k is odd and n is even

Construction 3: k and n are odd

Conclusion