Balanced edge colorings

Balanced edge colorings

This paper contains two principal results. The first proves that any graph G can be given a balanced proper edge coloring by k colors for any k ≥ χ′(G). Here balanced means that the number of vertices incident with any set of d colors is essentially fixed for each d, that is, for two different d-sets of colors the number of vertices incident with each of them can differ by at most 2. The second...

Multiply balanced edge colorings of multigraphs

In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out among the vertices in ways that are fair with respect to several notions of balance (such as between pairs of vertices, degrees of vertices in the both grap...

Balanced k-Colorings

While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest diierence between the size of any two color classes in that subset. The discrepancy is the minimum possible imbal...

Proper Bounded Edge-Colorings

For xed integers k 2, and for n-element sets X and colorings : X] k ?! f0; 1; : : :g where every color class is a matching and has cardinality at most u, we show that there exists a totally multicolored subset Y X with jY j max (c 1 n k u 1 2k?1 ; c 2 n k u 1 2k?1 ln u p n 1 2k?1) where c 1 ; c 2 > 0 are constants. This lower bound is tight up to constant factors for u = (n 1=2+) for every > 0....

Edge Colorings in Bipartite Graphs