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Signed k-independence in graphs

100%

Volkmann L.

Open Mathematics

2014

tom 12

nr 3

517-528

Let k ≥ 2 be an integer. A function f: V(G) → {−1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v] f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α sk(G) of G. In this work, we mainly present upper bounds on α sk (G), as for example α sk(G) ≤ n − 2⌈(Δ(G) + 2 − k)/2⌉, and we prove the Nordhaus-Gaddum type inequality $$\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$$, where n is the order, Δ(G) the maximum degree and $$\bar G$$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.

A lower bound for the packing chromatic number of the Cartesian product of cycles

75%

Jacobs Y., Jonck E., Joubert E.

Open Mathematics

2013

tom 11

nr 7

1344-1357

Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X i ⊆ V(G) is called an i-packing if each two distinct vertices in X i are more than i apart. A packing colouring of G is a partition X = {X 1, X 2, …, X k} of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C 4 □ C q). We will also show that if t is a multiple of four, then χρ(C 4 □ C q) = 9.

Ograniczanie wyników

2 Open Mathematics

1 Jacobs Y.

1 Jonck E.

1 Joubert E.

1 Volkmann L.

1 2014

1 2013

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Signed k-independence in graphs

A lower bound for the packing chromatic number of the Cartesian product of cycles