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Graphs with disjoint dominating and paired-dominating sets

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Southey J., Henning M.
Open Mathematics
|
2010
|
tom 8
|
nr 3
459-467
EN
A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.
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On odd and semi-odd linear partitions of cubic graphs

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Fouquet J.-L., Thuillier H., Vanherpe J.-M., Wojda A.
Discussiones Mathematicae Graph Theory
|
2009
|
tom 29
|
nr 2
275-292
EN
A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. In this paper we consider linear partitions of cubic simple graphs for which it is well known that la(G) = 2. A linear partition $L = (L_B,L_R)$ is said to be odd whenever each path of $L_B ∪ L_R$ has odd length and semi-odd whenever each path of $L_B$ (or each path of $L_R$) has odd length. In [2] Aldred and Wormald showed that a cubic graph G is 3-edge colourable if and only if G has an odd linear partition. We give here more precise results and we study moreover relationships between semi-odd linear partitions and perfect matchings.
3
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Mácajová and Škoviera conjecture on cubic graphs

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Fouquet J.-L., Vanherpe J.-M.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 2
315-333
EN
A conjecture of Mácajová and Skoviera asserts that every bridgeless cubic graph has two perfect matchings whose intersection does not contain any odd edge cut. We prove this conjecture for graphs with few vertices and we give a stronger result for traceable graphs.
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