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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Spanning trees with many or few colors in edge-colored graphs

100%
Broersma H., Li X.
Discussiones Mathematicae Graph Theory
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1997
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tom 17
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nr 2
259-269
EN
Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard.
2
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Isomorphisms and traversability of directed path graphs

100%
Broersma H., Li X.
Discussiones Mathematicae Graph Theory
|
2002
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tom 22
|
nr 2
215-228
EN
The concept of a line digraph is generalized to that of a directed path graph. The directed path graph Pₖ(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k+1 vertices or form a directed cycle on k vertices in D. In this introductory paper several properties of P₃(D) are studied, in particular with respect to isomorphism and traversability. In our main results, we characterize all digraphs D with P₃(D) ≅ D, we show that P₃(D₁) ≅ P₃(D₂) "almost always" implies D₁ ≅ D₂, and we characterize all digraphs with Eulerian or Hamiltonian P₃-graphs.
3
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A σ₃ type condition for heavy cycles in weighted graphs

81%
Zhang S., Li X., Broersma H.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 2
159-166
EN
A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree $d^w(v)$ of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least m; 2. w(xz) = w(yz) for every vertex z ∈ N(x)∩N(y) with d(x,y) = 2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/3. This generalizes a theorem of Fournier and Fraisse on the existence of long cycles in k-connected unweighted graphs in the case k = 2. Our proof of the above result also suggests a new proof to the theorem of Fournier and Fraisse in the case k = 2.
4
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Heavy subgraph pairs for traceability of block-chains

81%
Li B., Broersma H., Zhang S.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 2
287-307
EN
A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o−1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks. Obviously, every traceable graph is a block-chain, but the reverse does not hold. In this paper we characterize all the pairs of connected o−1-heavy graphs that guarantee traceability of block-chains. Our main result is a common extension of earlier work on degree sum conditions, forbidden subgraph conditions and heavy subgraph conditions for traceability
5
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Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

81%
Broersma H., Marchal B., Paulusma D., Salman A.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 1
143-162
EN
We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and H is a proper vertex coloring V→ {1,2,...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number l of colors, for which such colorings V→ {1,2,...,l} exist, in the worst case is a factor times the chromatic number (for path, tree, matching and star backbones). We show here that for split graphs and matching or star backbones, l is at most a small additive constant (depending on λ) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on l than the previously known bounds.
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