This paper presents techniques for handling symmetries in integer linear programs where variables can take integer values, extending previous work dealing exclusively with binary variables. Orthogonal array construction and coloring problems are used as illustrations.

For a given induced hereditary property P, a P-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property P. We consider the effectiveness of on-line P-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound fo...

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k − 1)-regular but not k-regular. We prove this conjecture by showing that the equation Pk−1 i=1 2 2i−1xi = “ −1 + Pk−1 i=1 2 2i−1 ” x0 has this property. This conjecture i...

A linear equation is r-regular, if, for every r-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoićič conjectured that the equation x1 +2x2 + · · ·+2xn−1−2xn = 0, for any n > 2, has a degree of regularity of n−1, which would verify a conjecture of Rado from 1933. Rado’s conjecture has since been verified with a d...

Given a nite directed graph, a coloring of its edges turns the graph into a nite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected nite graph of a uniform outdegree (constant outdegree of any vertex) is k-...

4-methylimidazole (4-mei) is a simple nitrogen-containing heterocyclic compound, and recently classified as group 2b by international agency for research on cancer (iarc). caramel coloring is widely applied food coloring in many food products including coffee, carbonated beverages, beer and wine. in present study, ɣ-irradiation was applied to caramel color iii to reduce the level of 4-mei witho...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. A graph G is s-degenerated for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerated graph G has a total coloring with ∆ + 1 colors if the...

The total graph T (G) of a multigraph G has as its vertices the set of edges and vertices of G and has an edge between two vertices if their corresponding elements are either adjacent or incident in G. We show that if G has maximum degree ∆(G), then T (G) is (2∆(G) − 1)-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for ∆(G) > 3 w...

A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph G, denoted by χ s (G), is the minimum number of colours needed in any strong edge colouring of G. A graph is said to be chordless if there is no cycle in the g...

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. In [2] we claimed that our algorithm MCColor produces an optimal coloring for every Meyniel graph. But later we found a mistake in the proof and a couterexample to the optimality, which we present here. MCColor can still be used to find a stable set that intersects all maximal cliques of a Meyniel graph ...