In this paper, by the sorting method of vertices, it is obtained that the equitable chromatic number and the equitable chromatic threshold of the Cartesian products of fans with bipartite graphs.

An equitable colouring of a balanced G-design (X,B) is a map φ∶B → C such that ∣bi(x) − bj(x)∣ ≤ 1 for any x ∈ X and i, j, with i ≠ j, being bi(x) the number of blocks containing the vertex x and coloured with the colour i. A c-colouring is a colouring in which exactly c colours are used. A c-colouring of type s is a colourings in which, for every vertex x, all the blocks containing x are colou...

An equitable graph coloring is a proper vertex coloring, such that the size of the color classes differ by at most one. We present a flow-based scheme for generating pruning rules for enumerative algorithms such as DSATUR. The scheme models the extendability of a partial (equitable) coloring into an equitable coloring via network flows. Computational experiments show that significant reductions...

Abstract From Adam Smith’s 1776 classic “The Wealth of Nations” until modern times, the principle fairness has remained one crucial principles on which tax systems are to be based. However, even and budgetary far from being gender equitable. This chapter provides an analysis ways in taxation, as well subsequent spending resources collected therewith, influence equality within a household, at wo...

Dr Cristina Villalobos — Myles and Sylvia Aaronson endowed professor in the School of Mathematical Statistical Sciences at The University Texas Rio Grande Valley (UTRGV), Director Center Excellence STEM Education, Fellow American Society talks to Nature Computational Science about her work on empowering underrepresented groups education gives insights into United Nations Sustainable Development...

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most dn(G )/ke vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove tha...

We study several cardinal, and ordinal–valued functions that are relatives of Hanf numbers. Let κ be an infinite cardinality, and let T ⊆ Lκ+,ω be a theory of cardinality ≤ κ, and let γ be an ordinal ≥ κ. Consider (1) μT (γ, κ) := min{μ∗ : ∀φ ∈ L∞,ω , with rk(φ) < γ, if T has the (φ, μ∗)order property then there exists a formula φ ′ (x; y) ∈ Lκ+,ω , such that for every χ ≥ κ, T has the (φ , χ)-...

An equitable two-coloring of a hypergraph H = (V, E) is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with property B problem Radhakrishnan and Srinivasan proved that if H is a k-uniform hypergraph with maximum vertex degree ∆(H) satisfying ∆(H) c 2 k−1 √ k ln k (1) for some absolute constant c > 0, then H is 2-colorable. By using...