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1
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On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs

100%
Seamone B.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 2
207-214
EN
A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.
2
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Matchings Extend to Hamiltonian Cycles in 5-Cube

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Wang F., Zhao W.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
217-231
EN
Ruskey and Savage asked the following question: Does every matching in a hypercube Qn for n ≥ 2 extend to a Hamiltonian cycle of Qn? Fink confirmed that every perfect matching can be extended to a Hamiltonian cycle of Qn, thus solved Kreweras’ conjecture. Also, Fink pointed out that every matching can be extended to a Hamiltonian cycle of Qn for n ∈ {2, 3, 4}. In this paper, we prove that every matching in Q5 can be extended to a Hamiltonian cycle of Q5.
3
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Ring elements as sums of units

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Lanski C., Maróti A.
Open Mathematics
|
2009
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tom 7
|
nr 3
395-399
EN
In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.
4
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Addendum to “Ring elements as sums of units”

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Lanski C., Maróti A.
Open Mathematics
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2013
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tom 11
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nr 5
984-984
EN
We give a comment to Theorem 1.1 published in our paper “Ring elements as sums of units” [Cent. Eur. J. Math., 2009, 7(3), 395–399].
5
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Chvátal's Condition cannot hold for both a graph and its complement

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Kostochka A., West D.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 1
73-76
EN
Chvátal's Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d₁, ...,dₙ in nondecreasing order, i < n/2 implies that $d_i > i$ or $d_{n-i} ≥ n-i$. We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2.
6
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A note on maximal common subgraphs of the Dirac's family of graphs

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Bucko J., Mihók P., Saclé J.-F., Woźniak M.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 3
385-390
EN
Let 𝓕 ⁿ be a given set of unlabeled simple graphs of order n. A maximal common subgraph of the graphs of the set 𝓕 ⁿ is a common subgraph F of order n of each member of 𝓕 ⁿ, that is not properly contained in any larger common subgraph of each member of 𝓕 ⁿ. By well-known Dirac's Theorem, the Dirac's family 𝓓𝓕 ⁿ of the graphs of order n and minimum degree δ ≥ [n/2] has a maximal common subgraph containing Cₙ. In this note we study the problem of determining all maximal common subgraphs of the Dirac's family $𝓓 𝓕 ^{2n}$ for n ≥ 2.
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