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1
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Maximal k-independent sets in graphs

100%
Blidia M., Chellali M., Favaron O., Meddah N.
Discussiones Mathematicae Graph Theory
|
2008
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tom 28
|
nr 1
151-163
EN
A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted iₖ(G) and βₖ(G). We give some relations between βₖ(G) and $β_j(G)$ and between iₖ(G) and $i_j(G)$ for j ≠ k. We study two families of extremal graphs for the inequality i₂(G) ≤ i(G) + β(G). Finally we give an upper bound on i₂(G) and a lower bound when G is a cactus.
2
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Difference labelling of cacti

80%
Sonntag M.
Discussiones Mathematicae Graph Theory
|
2003
|
tom 23
|
nr 1
55-65
EN
A graph G is a difference graph iff there exists S ⊂ IN⁺ such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = {{i,j}:i,j ∈ V ∧ |i-j| ∈ V}. It is known that trees, cycles, complete graphs, the complete bipartite graphs $K_{n,n}$ and $K_{n,n-1}$, pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.
3
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Rainbow Connectivity of Cacti and of Some Infinite Digraphs

80%
Alva-Samos J., Montellano-Ballesteros J. J.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
nr 2
301-313
EN
An arc-coloured digraph D = (V,A) is said to be rainbow connected if for every pair {u, v} ⊆ V there is a directed uv-path all whose arcs have different colours and a directed vu-path all whose arcs have different colours. The minimum number of colours required to make the digraph D rainbow connected is called the rainbow connection number of D, denoted rc⃗ (D). A cactus is a digraph where each arc belongs to exactly one directed cycle. In this paper we give sharp upper and lower bounds for the rainbow connection number of a cactus and characterize those cacti whose rainbow connection number is equal to any of those bounds. Also, we calculate the rainbow con- nection numbers of some infinite digraphs and graphs, and present, for each n ≥ 6, a tournament of order n and rainbow connection number equal to 2.
4
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M2-Edge Colorings Of Cacti And Graph Joins

80%
Czap J., Šugerek P., Ivančo J.
Discussiones Mathematicae Graph Theory
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2016
|
tom 36
|
nr 1
59-69
EN
An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.
5
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L(2, 1)-Labelings of Some Families of Oriented Planar Graphs

51%
Sen S.
Discussiones Mathematicae Graph Theory
|
2014
|
tom 34
|
nr 1
31-48
EN
In this paper we determine, or give lower and upper bounds on, the 2-dipath and oriented L(2, 1)-span of the family of planar graphs, planar graphs with girth 5, 11, 16, partial k-trees, outerplanar graphs and cacti.
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