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1
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Rainbow Connection Number of Graphs with Diameter 3

100%
Li H., Li X., Sun Y.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
nr 1
141-154
EN
A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.
2
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Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least six

100%
Bu Y., Lih K.-W., Wang W.
Discussiones Mathematicae Graph Theory
|
2011
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tom 31
|
nr 3
429-439
EN
An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring o G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ'ₐ(G). We prove that χ'ₐ(G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges whose girth is at least 6. This gives new evidence to a conjecture proposed in [Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett., 15 (2002) 623-626.]
3
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Maximum Edge-Colorings Of Graphs

88%
Jendrol’ S., Vrbjarová M.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 1
117-125
EN
An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) $\chi _r^\prime (G)$ is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 $\chi _r^\prime (G) \le 3$ for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i $\chi _1^\prime (G) = i$ , i = 1, 2, 3 are characterized. The precise value of the r-maximum index, r ≥ 1, is determined for trees and complete graphs.
4
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Almost-Rainbow Edge-Colorings of Some Small Subgraphs

88%
Krop E., Krop I.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 4
771-784
EN
Let f(n, p, q) be the minimum number of colors necessary to color the edges of Kn so that every Kp is at least q-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdös and Gyárfás. We show that [...] , slightly improving the bound of Axenovich. We make small improvements on bounds of Erdös and Gyárfás by showing [...] and for all even n ≢ 1(mod 3), f(n, 4, 5) ≤ n− 1. For a complete bipartite graph G= Kn,n, we show an n-color construction to color the edges of G so that every C4 ⊆ G is colored by at least three colors. This improves the best known upper bound of Axenovich, Füredi, and Mubayi.
5
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Three edge-coloring conjectures

75%
Schelp R.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 1
173-182
EN
The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.
6
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Spanning trees with many or few colors in edge-colored graphs

63%
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.
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