The Maximum Size of an Edge Cut and Graph Homomorphisms ()
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, Erdös 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.
Share and Cite:
S. Fan, H. Lai and J. Zhou, "The Maximum Size of an Edge Cut and Graph Homomorphisms,"
Applied Mathematics, Vol. 2 No. 10, 2011, pp. 1263-1269. doi:
10.4236/am.2011.210176.
Conflicts of Interest
The authors declare no conflicts of interest.
References
[1]
|
A. J. Bondy and U. S. R. Murty, “Graph Theory with Applications,” American Elsevier, New York, 1976.
|
[2]
|
P. Erd?s, “Problems and Results in Graph Theory and Combinatorial Analysis,” Graph Theory and RelatedTopics, Academic Press, New York, 1979.
|
[3]
|
M. O. Albertson and K. L. Gibbons, “Ho-momorphisms of 3-Chromatic Graphs,” Discrete Mathe-matics, Vol. 54, No. 2, 1985, pp. 127-132.
doi:10.1016/0012-365X(85)90073-1
|
[4]
|
M. O. Albert-son, P. A. Catlin and L. Gibbons, “Homomorphisms of 3-Chromatic Graphs II,” Congressus Numerantium, Vol. 47, 1985, pp. 19-28.
|
[5]
|
P. A. Catlin, “Graph Homo-morphisms onto the 5-Cycle,” Journal of Combinatorial Theory, Series B, Vol. 45, No. 2, 1988, pp. 199-211.
doi:10.1016/0095-8956(88)90069-X
|
[6]
|
E. R. Schei-nerman and D. H. Ullman, “Fractional Graph Theory,” John Wiley Sons, Inc., New York, 1997.
|
[7]
|
G. G. Gao and X. X. Zhu, “Star-Extremal Graphs and the Lexico-graphic Product,” Discrete Mathematics, Vol. 153, 1996, pp. 147-156.
|
[8]
|
S. Poljak and Z. Tuza, “Maximum Bi-partite Subgraphs of Kneser Graphs,” Graphs and Com-binatorics, Vol. 3, 1987, pp. 191-199. doi:10.1007/BF01788540
|