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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Decompositions of multigraphs into parts with two edges

100%
Ivančo J., Meszka M., Skupień Z.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 1
113-121
EN
Given a family 𝓕 of multigraphs without isolated vertices, a multigraph M is called 𝓕-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of 𝓕. We present necessary and sufficient conditions for the existence of such decompositions if 𝓕 comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.
2
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Perfect Set of Euler Tours of Kp,p,p

100%
Govindan T., Muthusamy A.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 4
783-796
EN
Bermond conjectured that if G is Hamilton cycle decomposable, then L(G), the line graph of G, is Hamilton cycle decomposable. In this paper, we construct a perfect set of Euler tours for the complete tripartite graph Kp,p,p for any prime p and hence prove Bermond’s conjecture for G = Kp,p,p.
3
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Characterization of Line-Consistent Signed Graphs

100%
Slilaty D. C., Zaslavsky T.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 3
589-594
EN
The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede’s relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede’s theorem as well as a structural description of line-consistent signed graphs.
4
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The structure and existence of 2-factors in iterated line graphs

100%
Ferrara M., Gould R., Hartke S.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 3
507-526
EN
We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where $L^k(G)$ contains a 2-factor.
5
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Arithmetically maximal independent sets in infinite graphs

100%
Bylka S.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 1-2
167-182
EN
Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed. A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.
6
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Characterizations of planar plick graphs

100%
Kulli V., Basavanagoud B.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 1
41-45
EN
In this paper we present characterizations of graphs whose plick graphs are planar, outerplanar and minimally nonouterplanar.
7
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On •-Line Signed Graphs L•(S)

75%
Sinha D., Dhama A.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 2
215-227
EN
A signed graph (or sigraph for short) is an ordered pair S = (Su,σ), where Su is a graph, G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a sigraph S its •-line sigraph, L•(S) is the sigraph in which the edges of S are represented as vertices, two of these vertices are defined adjacent whenever the corresponding edges in S have a vertex in common, any such L-edge ee′ has the sign given by the product of the signs of the edges incident with the vertex in e ∩ e′. In this paper we establish a structural characterization of •-line sigraphs, extending a well known characterization of line graphs due to Harary. Further we study several standard properties of •-line sigraphs, such as the balanced •-line sigraphs, sign-compatible •-line sigraphs and C-sign-compatible •-line sigraphs.
8
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Sum List Edge Colorings of Graphs

75%
Kemnitz A., Marangio M., Voigt M.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 3
709-722
EN
Let G = (V,E) be a simple graph and for every edge e ∈ E let L(e) be a set (list) of available colors. The graph G is called L-edge colorable if there is a proper edge coloring c of G with c(e) ∈ L(e) for all e ∈ E. A function f : E → ℕ is called an edge choice function of G and G is said to be f-edge choosable if G is L-edge colorable for every list assignment L with |L(e)| = f(e) for all e ∈ E. Set size(f) = ∑e∈E f(e) and define the sum choice index χ′sc(G) as the minimum of size(f) over all edge choice functions f of G. There exists a greedy coloring of the edges of G which leads to the upper bound χ′sc(G) ≤ 1/2 ∑v∈V d(v)2. A graph is called sec-greedy if its sum choice index equals this upper bound. We present some general results on the sum choice index of graphs including a lower bound and we determine this index for several classes of graphs. Moreover, we present classes of sec-greedy graphs as well as all such graphs of order at most 5.
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