The Maximum Size of an Edge Cut and Graph Homomorphisms

Hong‐Jian Lai ,Fan Suo-Hai ,Ju Zhou

doi:10.4236/am.2011.210176

Abstract

For a graph G, let b(G)=max﹛|D|: Dis an edge cut of G﹜ . For graphs G and H, a map Ψ: V(G)→V(H) is a graph homomorphism if for each e=uv∈E(G), Ψ(u)Ψ(v)∈E(H). In 1979, Erdos proved by probabilistic methods that for p ≥ 2 with if there is a graph homomorphism from G onto Kp then b(G)≥f(p)|E(G)| In this paper, we obtained the best possible lower bounds of b(G) for graphs G with a graph homomorphism onto a Kneser graph or a circulant graph and we characterized the graphs G reaching the lower bounds when G is an edge maximal graph with a graph homomorphism onto a complete graph, or onto an odd cycle.

Highlights

In this paper, the graphs we consider are finite, simple and connected
In 1979, Erdös proved by probabilistic methods that for p ≥ 2 with f
In Subsection 3.1, we give an alternative proof of Theorem 1.1 and characterize the graphs G reaching the lower bound in Theorem 1.1 when G is edge-maximal Kp-colorable

Summary

Introduction

The graphs we consider are finite, simple and connected. Undefined notation and terminology will follow those in [1]. De fSin,eS for some b G max D : D is an edge cut of G. plete graph with p vertices and C2 p 1 to denote an odd cycle with V C2 p 1 vi , i Z2 p 1 and. In Subsection 3.1, we give an alternative proof of Theorem 1.1 and characterize the graphs G reaching the lower bound in Theorem 1.1 when G is edge-maximal Kp-colorable. In Subsection 3.3, we show the validity of Theorem 1.3 and Theorem 1.4 and characterize the graphs G reaching the lower bound in Theorem 1.4 when G is edge-maximal K p/q -colorable. In Subsection 3.5, we show a best possible lower bound for b G when G has a graph homomorphism onto an odd cycle C2 p 1 and characterize the graphs reaching the lower bound among all edge-maximal C2 p 1 -colorable graphs. There are a lot of researches about graph homomorphism can be found in [3,4,5,6,7]

An Associate Theorem

E H and k i 1

Graphs with a Graph Homomorphism onto a Complete Graph

D 1 X 1 Y

Graphs with a Graph Homomorphism onto a Kneser Graph

Graphs with a Graph Homomorphism onto a Circulant Graph

Graphs with a Graph Homomorphism onto an Odd Cycle