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Wiener index of generalized stars and their quadratic line graphs

100%
Dobrynin A., Mel'nikov L.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 1
161-175
EN
The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property are constructed.
2
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4-chromatic Koester graphs

100%
Dobrynin A., Mel'nikov L.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 4
617-627
EN
Let G be a simple 4-regular plane graph and let S be a decomposition of G into edge-disjoint cycles. Suppose that every two adjacent edges on a face belong to different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Studies of coloring of graphs of this kind were originated by Grötzsch. Two 4-chromatic graphs generated by circles in the plane were constructed by Koester in 1984 [10,11,12]. Until now, no other examples of such graphs were known. We present fourteen new 4-chromatic graphs generated by circles in the plane.
3
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Some news about oblique graphs

81%
Dobrynin A., Melnikov L., Schreyer J., Walther H.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 1
39-50
EN
A k-gon α of a polyhedral graph G(V,E,F) is of type ⟨b₁,b₂,...,bₖ⟩ if the vertices incident with α in cyclic order have degrees b₁,b₂,...,bₖ and ⟨b₁,b₂,...,bₖ⟩ is the lexicographic minimum of all such sequences available for α. A polyhedral graph G is oblique if it has no two faces of the same type. Among others it is shown that an oblique graph contains vertices of degree 3.
4
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Erdős regular graphs of even degree

81%
Dobrynin A., Mel'nikov L., Pyatkin A.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 2
269-279
EN
In 1960, Dirac put forward the conjecture that r-connected 4-critical graphs exist for every r ≥ 3. In 1989, Erdös conjectured that for every r ≥ 3 there exist r-regular 4-critical graphs. A method for finding r-regular 4-critical graphs and the numbers of such graphs for r ≤ 10 have been reported in [6,7]. Results of a computer search for graphs of degree r = 12,14,16 are presented. All the graphs found are both r-regular and r-connected.
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