An Upper Bound on the Radius of a 3-Vertex-Connected C4-Free Graph

Blessings T Fundikwa,Jaya P Mazorodze,Simon Mukwembi

doi:10.1155/2020/8367408

Blessings T Fundikwa, Jaya P Mazorodze + Show 1 more

Open Access

https://doi.org/10.1155/2020/8367408

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Abstract

We show that if G is a 3-vertex-connected C4-free graph of order n and radius r, then the inequality r≤2n/9+O1 holds. Moreover, graphs are constructed to show that the bounds are asymptotically sharp.

Highlights

E eccentricity ec(v) of a vertex v ∈ V is the maximum distance between v and any other vertex in G. e value of the minimum eccentricity of the vertices of G is called the radius of G denoted by rad(G). e degree deg(v) of a vertex v of G is the number of edges incident with v. e minimum degree δ(G) is the minimum of the degrees of vertices in G
E open neighbourhood N(v) of a vertex v is the set of all vertices of G adjacent to v. e closed neighbourhood N[v] of v is the set N(v) ∪ {v}
We are concerned with upper bounds on radius in terms of order and each of the three classical connectivity measures, namely, minimum degree, edgeconnectivity, and vertex-connectivity for general graphs and for graphs with forbidden subgraphs such as C3 and C4

Summary

Introduction

Let G (V, E) be a finite, connected, undirected graph with vertex set V and edge set E. e distance dG(u, v) between two vertices u, v of G is the length of a shortest u-v path in G. Erdős et al [2] and independently Dankelmann et al [3] proved that if G is a connected graph of order n and minimum degree δ ≥ 2, rad(G). In [5], showed that if G is a triangle-free 3-edge-connected graph, rad(G). Bounds on the radius in terms of Journal of Mathematics order and vertex-connectivity were given by Harant and Walther [6]. It is worth noting that, while it has not been formally communicated, the bounds on the radius of triangle-free graphs with respect to order and vertex-connectivity are exactly the same as those for graphs without restrictions where subgraphs are concerned. There are no known bounds reported on the radius of C4-free graphs with respect to order and vertex-connectivity. Note that CkS is a 3-vertex-connected C4-free graph and that whenever k ≡ 0 mod, we have

Results

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