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1
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On the stability for pancyclicity

100%
Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2001
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tom 21
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nr 2
223-228
EN
A property P defined on all graphs of order n is said to be k-stable if for any graph of order n that does not satisfy P, the fact that uv is not an edge of G and that G + uv satisfies P implies $d_G(u) + d_G(v) < k$. Every property is (2n-3)-stable and every k-stable property is (k+1)-stable. We denote by s(P) the smallest integer k such that P is k-stable and call it the stability of P. This number usually depends on n and is at most 2n-3. A graph of order n is said to be pancyclic if it contains cycles of all lengths from 3 to n. We show that the stability s(P) for the graph property "G is pancyclic" satisfies max(⎡6n/5]⎤-5, n+t) ≤ s(P) ≤ max(⎡4n/3]⎤-2,n+t), where t = 2⎡(n+1)/2]⎤-(n+1).
2
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Problems remaining NP-complete for sparse or dense graphs

100%
Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
1995
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tom 15
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nr 1
33-41
EN
For each fixed pair α,c > 0 let INDEPENDENT SET ($m ≤ cn^α$) and INDEPENDENT SET ($m ≥ (ⁿ₂) - cn^α$) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m ≤ cn^α$ or $m ≥ (ⁿ₂) - cn^α$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ($m ≤ n + cn^α$) and HAMILTONIAN PATH ($m ≤ n + cn^α$) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m ≤ n + cn^α$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with m ≥ (1 - ϵ)(ⁿ₂) edges. We prove that these six restricted problems remain NP-complete. Finally, we consider sufficient conditions for a graph to have a Hamiltonian circuit. These conditions are based on degree sums and neighborhood unions of independent vertices, respectively. Lowering the required bounds the problem HAMILTONIAN CIRCUIT jumps from 'easy' to 'NP-complete'.
3
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A new upper bound for the chromatic number of a graph

100%
Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
137-142
EN
Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ [(n+ω+1-α)/2]. Moreover, χ(G) ≤ [(n+ω-α)/2], if either ω + α = n + 1 and G is not a split graph or α + ω = n - 1 and G contains no induced $K_{ω+3}- C₅$.
4
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Bounds for the rainbow connection number of graphs

100%
Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2011
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tom 31
|
nr 2
387-395
EN
An edge-coloured graph G is rainbow-connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow-connected. In this paper we show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.
5
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The cycle-complete graph Ramsey number r(C₅,K₇)

100%
Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2005
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tom 25
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nr 1-2
129-139
EN
The cycle-complete graph Ramsey number r(Cₘ,Kₙ) is the smallest integer N such that every graph G of order N contains a cycle Cₘ on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r(Cₘ,Kₙ) = (m-1)(n-1)+1 for all m ≥ n ≥ 3 (except r(C₃,K₃) = 6). This conjecture holds for 3 ≤ n ≤ 6. In this paper we will present a proof for r(C₅,K₇) = 25.
6
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The flower conjecture in special classes of graphs

64%
Ryjáček Z., Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
1995
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tom 15
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nr 2
179-184
EN
We say that a spanning eulerian subgraph F ⊂ G is a flower in a graph G if there is a vertex u ∈ V(G) (called the center of F) such that all vertices of G except u are of the degree exactly 2 in F. A graph G has the flower property if every vertex of G is a center of a flower. Kaneko conjectured that G has the flower property if and only if G is hamiltonian. In the present paper we prove this conjecture in several special classes of graphs, among others in squares and in a certain subclass of claw-free graphs.
7
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Colouring graphs with prescribed induced cycle lengths

64%
Randerath B., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 2
267-281
EN
In this paper we study the chromatic number of graphs with two prescribed induced cycle lengths. It is due to Sumner that triangle-free and P₅-free or triangle-free, P₆-free and C₆-free graphs are 3-colourable. A canonical extension of these graph classes is $𝓖^I(4,5)$, the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of $𝓖^I(4,5)$ are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind, i.e., we have polynomial time algorithms to 3-colour every $G ∈ 𝓖^I(n₁,n₂)$ with n₁,n₂ ≥ 4 (see Table 1). Furthermore, we consider the related problem of finding a χ-binding function for the class $𝓖^I(n₁,n₂)$. Here we obtain the surprising result that there exists no linear χ-binding function for $𝓖^I(3,4)$.
8
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On the computational complexity of (O,P)-partition problems

64%
Kratochvíl J., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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1997
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tom 17
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nr 2
253-258
EN
We prove that for any additive hereditary property P > O, it is NP-hard to decide if a given graph G allows a vertex partition V(G) = A∪B such that G[A] ∈ 𝓞 (i.e., A is independent) and G[B] ∈ P.
9
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New sufficient conditions for hamiltonian and pancyclic graphs

64%
Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
29-38
EN
For a graph G of order n we consider the unique partition of its vertex set V(G) = A ∪ B with A = {v ∈ V(G): d(v) ≥ n/2} and B = {v ∈ V(G):d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.
10
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A lower bound on the independence number of a graph in terms of degrees

64%
Harant J., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 3
431-437
EN
For a connected and non-complete graph, a new lower bound on its independence number is proved. It is shown that this bound is realizable by the well known efficient algorithm MIN.
11
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Graphs with rainbow connection number two

64%
Kemnitz A., Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2011
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tom 31
|
nr 2
313-320
EN
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that rc(G) = 2 for every connected graph G of order n and size m, where $\binom{n-1}{2} + 1 ≤ m ≤ \binom{n}{2} - 1$. We also characterize graphs with rainbow connection number two and large clique number.
12
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On Maximum Weight of a Bipartite Graph of Given Order and Size

52%
Horňák M., Jendrol’ S., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
147-165
EN
The weight of an edge xy of a graph is defined to be the sum of degrees of the vertices x and y. The weight of a graph G is the minimum of weights of edges of G. More than twenty years ago Erd˝os was interested in finding the maximum weight of a graph with n vertices and m edges. This paper presents a complete solution of a modification of the above problem in which a graph is required to be bipartite. It is shown that there is a function w*(n,m) such that the optimum weight is either w*(n,m) or w*(n,m) + 1.
13
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Rainbow Connection Number of Dense Graphs

52%
Li X., Liu M., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
|
nr 3
603-611
EN
An edge-colored graph G is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we show that rc(G) ≤ 3 if |E(G)| ≥ [...] + 2, and rc(G) ≤ 4 if |E(G)| ≥ [...] + 3. These bounds are sharp.
14
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The Ramsey number r(C₇,C₇,C₇)

52%
Faudree R., Schelten A., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2003
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tom 23
|
nr 1
141-158
EN
Bondy and Erdős [2] have conjectured that the Ramsey number for three cycles Cₖ of odd length has value r(Cₖ,Cₖ,Cₖ) = 4k-3. We give a proof that r(C₇,C₇,C₇) = 25 without using any computer support.
15
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Rainbow Connection In Sparse Graphs

52%
Kemnitz A., Przybyło J., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
|
2013
|
tom 33
|
nr 1
181-192
EN
An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.
16
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Preface

52%
Harant J., Voigt M., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2002
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tom 22
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nr 1
5-6
17
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On Sequential Heuristic Methods for the Maximum Independent Set Problem

52%
Lê N. C., Brause C., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
nr 2
415-426
EN
We consider sequential heuristics methods for the Maximum Independent Set (MIS) problem. Three classical algorithms, VO [11], MIN [12], or MAX [6] , are revisited. We combine Algorithm MIN with the α-redundant vertex technique[3]. Induced forbidden subgraph sets, under which the algorithms give maximum independent sets, are described. The Caro-Wei bound [4,14] is verified and performance of the algorithms on some special graphs is considered.
18
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Graphs with 3-Rainbow Index n − 1 and n − 2

52%
Li X., Schiermeyer I., Yang K., Zhao Y.
Discussiones Mathematicae Graph Theory
|
2015
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tom 35
|
nr 1
105-120
EN
Let G = (V (G),E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G), where k is an integer such that 2 ≤ k ≤ n. Chartrand et al. got that the k-rainbow index of a tree is n−1 and the k-rainbow index of a unicyclic graph is n−1 or n−2. So there is an intriguing problem: Characterize graphs with the k-rainbow index n − 1 and n − 2. In this paper, we focus on k = 3, and characterize the graphs whose 3-rainbow index is n − 1 and n − 2, respectively.
19
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Graphs with 4-Rainbow Index 3 and n − 1

52%
Li X., Schiermeyer I., Yang K., Zhao Y.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 2
387-398
EN
Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for every set S of k vertices of V (G) is called the k-rainbow index of G, denoted by rxk(G). Notice that a lower bound and an upper bound of the k-rainbow index of a graph with order n is k − 1 and n − 1, respectively. Chartrand et al. got that the k-rainbow index of a tree with order n is n − 1 and the k-rainbow index of a unicyclic graph with order n is n − 1 or n − 2. Li and Sun raised the open problem of characterizing the graphs of order n with rxk(G) = n − 1 for k ≥ 3. In early papers we characterized the graphs of order n with 3-rainbow index 2 and n − 1. In this paper, we focus on k = 4, and characterize the graphs of order n with 4-rainbow index 3 and n − 1, respectively.
20
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Extending the MAX Algorithm for Maximum Independent Set

52%
Lê N. C., Brause C., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 2
365-386
EN
The maximum independent set problem is an NP-hard problem. In this paper, we consider Algorithm MAX, which is a polynomial time algorithm for finding a maximal independent set in a graph G. We present a set of forbidden induced subgraphs such that Algorithm MAX always results in finding a maximum independent set of G. We also describe two modifications of Algorithm MAX and sets of forbidden induced subgraphs for the new algorithms.
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