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