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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Destroying symmetry by orienting edges: complete graphs and complete bigraphs

100%
Harary F., Jacobson M.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 2
149-158
EN
Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group. We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs $K_{s,t}$, s ≤ t, this number does not always exist. We determine for s ≤ 4 the values of t for which this number does exist.
2
Content available remote

An Oriented Version of the 1-2-3 Conjecture

80%
Baudon O., Bensmail J., Sopena É.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 1
141-156
EN
The well-known 1-2-3 Conjecture addressed by Karoński, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1, 2, 3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ receive distinct sums of outgoing weights. This result is tight in the sense that some oriented graphs do not admit such an assignment using the weights from {1, 2} only. We finally prove that deciding whether two weights are sufficient for a given oriented graph is an NP-complete problem. These results also hold for product or list versions of this problem.
3
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Independent Detour Transversals in 3-Deficient Digraphs

70%
van Aardt S., Frick M., Singleton J.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 2
261-275
EN
In 1982 Laborde, Payan and Xuong [Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173-177 (Teubner-Texte Math., 59 1983)] conjectured that every digraph has an independent detour transversal (IDT), i.e. an independent set which intersects every longest path. Havet [Stable set meeting every longest path, Discrete Math. 289 (2004) 169-173] showed that the conjecture holds for digraphs with independence number two. A digraph is p-deficient if its order is exactly p more than the order of its longest paths. It follows easily from Havet’s result that for p = 1, 2 every p-deficient digraph has an independent detour transversal. This paper explores the existence of independent detour transversals in 3-deficient digraphs.
4
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Every 8-Traceable Oriented Graph Is Traceable

70%
Aardt S. A. v.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 4
963-973
EN
A digraph of order n is k-traceable if n ≥ k and each of its induced subdigraphs of order k is traceable. It is known that if 2 ≤ k ≤ 6, every k-traceable oriented graph is traceable but for k = 7 and for each k ≥ 9, there exist k-traceable oriented graphs that are nontraceable. We show that every 8-traceable oriented graph is traceable.
5
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Signed domination and signed domatic numbers of digraphs

70%
Volkmann L.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 3
415-427
EN
Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → {-1,1} be a two-valued function. If $∑_{x ∈ N¯[v]}f(x) ≥ 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number $γ_S(D)$ of D. A set ${f₁,f₂,...,f_d}$ of signed dominating functions on D with the property that $∑_{i = 1}^d f_i(x) ≤ 1$ for each x ∈ V(D), is called a signed dominating family (of functions) on D. The maximum number of functions in a signed dominating family on D is the signed domatic number of D, denoted by $d_S(D)$. In this work we show that $4-n ≤ γ_S(D) ≤ n$ for each digraph D of order n ≥ 2, and we characterize the digraphs attending the lower bound as well as the upper bound. Furthermore, we prove that $γ_S(D) + d_S(D) ≤ n + 1$ for any digraph D of order n, and we characterize the digraphs D with $γ_S(D) + d_S(D) = n + 1$. Some of our theorems imply well-known results on the signed domination number of graphs.
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