OJDM  Vol.4 No.4 , October 2014
Balance in Random Trees
We prove that a random labeled (unlabeled) tree is balanced. We also prove that random labeled and unlabeled trees are strongly k-balanced for any k ≥ 3. Definition: Color the vertices of graph G with two colors. Color an edge with the color of its endpoints if they are colored with the same color. Edges with different colored endpoints are left uncolored. G is said to be balanced if neither the number of vertices nor and the number of edges of the two different colors differs by more than one.

Cite this paper
Akhmedov, A. and Shreve, W. (2014) Balance in Random Trees. Open Journal of Discrete Mathematics, 4, 97-108. doi: 10.4236/ojdm.2014.44013.
[1]   Lee, S.-M., Liu, A. and Tan, S.K. (1992) On Balanced Graphs. Congressus Numerantium, 87, 59-64.

[2]   Cahit, I. (1987) Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs, Ars Combinatoria, 23, 201-207.

[3]   Gallian, J.A. (2009) A Dynamical Survey of Graph Labeling. The Electronics Journal of Combinatorics, Dynamic Survey 6, 43 p. (electronic).

[4]   Graham, R. and Sloane, N. (2009) On Additive Bases and Harmonious Graphs. SIAM Journal of Algebraic and Discrete Mathematics, 1, x382-x404.

[5]   Cahit, I. (1990) On Cordial and 3-Equitable Graphs, Utilitas Mathematica, 37, 189-198.

[6]   Cayley, A. (1889) A Theorem on Trees. The Quarterly Journal of Mathematics, 23, 376-378.

[7]   West, D.B. (2001) Introduction to Graph Theory. 2nd Edition, Prentice-Hall, Inc., Upper Saddle River, 82-83.

[8]   Otter, R. (1948) The Number of Trees. Annals of Mathematics, 49, 583-599.

[9]   Bollobás, B. and Guy, R. (1983) Equitable and Proportional Coloring of Trees. Journal of Combinatorial Theory, Series B, 34, 177-186.

[10]   Ben-Eliezer, I. and Krivelevich, M. (2009) Perfectly Balanced Partitions of Smoothed Graphs. Electronic Journal of Combinatorics, 16, Note N14.

[11]   Bollobás, B. (2001) Random Graphs. Cambridge Studies in Advanced Mathematics (Book 73). 2nd Edition, Cambridge University Press, Cambridge.

[12]   Goh, W. and Schmutz, E. (1994) Unlabeled Trees: Distribution of the Maximum Degree. Random Structures and Algorithms, 5, 13-24.

[13]   Kong, M.C., Lee, S.-M., Seah, E. and Tang, A. (2008) A Complete Characterization of Balanced Graphs (English Summary). Journal of Combinatorial Mathematics and Combinatorial Computing, 66, 225-236.

[14]   Moon, J.W. (1968) On the Maximum Degree in a Random Tree. The Michigan Mathematical Journal, 15, 429-432.

[15]   Rényi, A. (1959) Some Remarks on the Theory of Trees. Magyar Tud. Akad. Mat. Kutat Int. Kzl, 4, 73-85.

[16]   Meir, A. and Moon, J.W. (1968) On Nodes of Degree Two in Random Trees. Mathematika, 15, 188-192.

[17]   Drmota, M. and Gittenberger, B. (1999) Distribution of Nodes of Given Degree in Random Trees. Journal of Graph Theory, 31, 227-253.