For any simple graph G, let f be a labeling of the verices of G by Z3 = {0,1,2}. Define f as above. We say f is 3-equita

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For Any Simple Graph G Let F Be A Labeling Of The Verices Of G By Z3 0 1 2 Define F As Above We Say F Is 3 Equita 1 (41.1 KiB) Viewed 30 times

For any simple graph G, let f be a labeling of the verices of G by Z3 = {0,1,2}. Define f as above. We say f is 3-equitable if |vf() – vf(i) <1 and Jef(i) - ef(i) <1, for i + j, i, j = {0,1,2}. This definition can be generalized to k – equitable for other values of k similarly. It is not known is all trees are k-equitable for k > 4. 2. Prove the statement: If the vertices of a tree are properly colored black and white and there are more black vertices than white vertices, then there is at least one end-vertex colored black. Note: A proper vertex coloring requires adjacent vertices to be of different colors.