A subset D of V (G) is called an equitable dominating set of a graph G if for every v ∈ (V − D), there exists a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| 6 1. The minimum cardinality of such a dominating set is denoted by γe(G) and is called equitable domination number of G. In this paper we introduce the equitable edge domination and equitable edge domatic number in a graph, exact...

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...

Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts, such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is dn/re or bn/rc. We prove that for all a ≥ 1, if the maximum degree of H satisfies ∆(H) ≤ k then H has an equitable coloring with k a ln k (1 − ok(1)) parts. In particul...

With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishin...

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al. [1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not appro...

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...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,.... | V | } jg such that if each edge uv is assigned the label -1 if |[f (u)]2 [f (v)]2 | = -1(mod 4), the label 0 if |[f (u)]2 [f (v)]2 | = 0(mod 4) and the label 1 if |[f (u)]2 [f (v)]| = 1(mod 4), then the number of edges labeled with i and the number of edges labelled with j differ by a...