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An equitable k-coloring of a graph G is a proper k-coloring, for which any two color classes differ in size by at most one. Equitable colorings naturally arise in some scheduling, partitioning, and load balancing problems [1, 15, 16]. Pemmaraju [13] and Janson and Ruciński [6] used equitable colorings to derive deviation bounds for sums of dependent random variables that exhibit limited depende...

A balanced coloring of a graph G means a triple {P1, P2,X} of mutually disjoint subsets of the vertex-set V (G) such that V (G) = P1⊎P2⊎X and |P1| = |P2|. A balanced decomposition associated with the balanced coloring V (G) = P1⊎P2⊎X of G is defined as a partition of V (G) = V1⊎· · ·⊎Vr (for some r) such that, for every i ∈ {1, · · · , r}, the subgraphG[Vi] of G is connected and |Vi∩P1| = |Vi∩P...

Let G be a graph and let S = (s1,s2,..., sk) non-decreasing sequence of positive integers. An S-packing coloring is mapping c : V(G) ? {1, 2,..., k} with the following property: if c(u) c(v) i, then d(u,v) > si for any i 2,...,k}. In particular, (1, 2, 3, ..., k), well known under name packing coloring. Next, r integer u,v V(G). A vertex u r-distance dominates v dG(u, v)? r. this paper, we p...

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph Kn that has no rainbow spanning subgraph with diameter at most t. We prove s(n, t) = (n−2 2 ) +1 for n, t ≥ 3, while s(n, 2) = (n−2 2 )

Abstract. Call the set S1 × · · · × St t–dimensional m–box if |Si| = m for every i = 1, . . . , t. Let Rt(m, r) be the smallest integer R such that for every r–coloring of t–fold cartesian product of [R] one can find a monochromatic t–dimensional m–box. We give a lower and an upper bound for Rt(m, r). We also consider the discrepancy problem connected to this set-system. Among other bounds we p...

A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex coloring of trees arises in areas such as phylogenetics, linguistics, etc. eg, a perfect phylogenetic tree is one in which the states of each character induce a...

Let f be a nonnegative integer valued function on the vertex set of graph. A graph is strictly f-degenerate if each nonempty subgraph ? has v such that deg?(v)<f(v). In this paper, we define new concept, transversal, which generalizes list coloring, signed DP-coloring, L-forested-coloring, and (f1,f2,…,fs)-partition. cover G H with V(H)=?v?V(G)Xv, where Xv={(v,1),(v,2),…,(v,s)}; edge M=?uv?E(G)...