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1
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Hamilton cycles in almost distance-hereditary graphs

100%
Chen B., Ning B.
Open Mathematics
|
2016
|
tom 14
|
nr 1
19-28
EN
Let G be a graph on n ≥ 3 vertices. A graph G is almost distance-hereditary if each connected induced subgraph H of G has the property dH(x, y) ≤ dG(x, y) + 1 for any pair of vertices x, y ∈ V(H). Adopting the terminology introduced by Broersma et al. and Čada, a graph G is called 1-heavy if at least one of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has degree at least n/2, and is called claw-heavy if each claw of G has a pair of end vertices with degree sum at least n. In this paper we prove the following two theorems: (1) Every 2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian. (2) Every 3-connected, 1-heavy and almost distance-hereditary graph is Hamiltonian. The first result improves a previous theorem of Feng and Guo [J.-F. Feng and Y.-B. Guo, Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws, Optimazation 57 (2008), no. 1, 135–141]. For the second result, its connectedness condition is sharp since Feng and Guo constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian.
2
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Forbidden Pairs and (k,m)-Pancyclicity

100%
Crane C. B.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
649-663
EN
A graph G on n vertices is said to be (k, m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each r ∈ {m, m+1, . . . , n}. This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree, Gould, Jacobson, and Lesniak in 2004. The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. We consider pairs of subgraphs that, when forbidden, guarantee hamiltonicity for 2-connected graphs on n ≥ 10 vertices. There are exactly ten such pairs. For each integer k ≥ 1 and each of eight such subgraph pairs {R, S}, we determine the smallest value m such that any 2-connected {R, S}-free graph on n ≥ 10 vertices is guaranteed to be (k,m)-pancyclic. Examples are provided that show the given values are best possible. Each such example we provide represents an infinite family of graphs.
3
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Heavy Subgraphs, Stability and Hamiltonicity

100%
Li B., Ning B.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
691-710
EN
Let G be a graph. Adopting the terminology of Broersma et al. and Čada, respectively, we say that G is 2-heavy if every induced claw (K1,3) of G contains two end-vertices each one has degree at least |V (G)|/2; and G is o-heavy if every induced claw of G contains two end-vertices with degree sum at least |V (G)| in G. In this paper, we introduce a new concept, and say that G is S-c-heavy if for a given graph S and every induced subgraph G′ of G isomorphic to S and every maximal clique C of G′, every non-trivial component of G′ − C contains a vertex of degree at least |V (G)|/2 in G. Our original motivation is a theorem of Hu from 1999 that can be stated, in terms of this concept, as every 2-connected 2-heavy and N-c-heavy graph is hamiltonian, where N is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs S such that every 2-connected o-heavy and S-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu’s theorem. Furthermore, our main result improves or extends several previous results.
4
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The Dichromatic Number of Infinite Families of Circulant Tournaments

100%
Javier N., Llano B.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 1
221-238
EN
The dichromatic number dc(D) of a digraph D is defined to be the minimum number of colors such that the vertices of D can be colored in such a way that every chromatic class induces an acyclic subdigraph in D. The cyclic circulant tournament is denoted by [...] T=C→2n+1(1,2,…,n) $T = \overrightarrow C _{2n + 1} (1,2, \ldots ,n)$ , where V (T) = ℤ2n+1 and for every jump j ∈ {1, 2, . . . , n} there exist the arcs (a, a + j) for every a ∈ ℤ2n+1. Consider the circulant tournament [...] C→2n+1〈k〉 $\overrightarrow C _{2n + 1} \left\langle k \right\rangle $ obtained from the cyclic tournament by reversing one of its jumps, that is, [...] C→2n+1 〈k〉 $\overrightarrow C _{2n + 1} \left\langle k \right\rangle $ has the same arc set as [...] C→2n+1(1,2,…,n) $\overrightarrow C _{2n + 1} (1,2, \ldots ,n)$ except for j = k in which case, the arcs are (a, a − k) for every a ∈ ℤ2n+1. In this paper, we prove that [...] dc(C→2n+1 〈k〉)∈{2,3,4} $dc ( {\overrightarrow C _{2n + 1} \left\langle k \right\rangle } ) \in \{ 2,3,4\}$ for every k ∈ {1, 2, . . . , n}. Moreover, we classify which circulant tournaments [...] C→2n+1 〈k〉 $\overrightarrow C _{2n + 1} \left\langle k \right\rangle$ are vertex-critical r-dichromatic for every k ∈ {1, 2, . . . , n} and r ∈ {2, 3, 4}. Some previous results by Neumann-Lara are generalized.
5
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Onq-Power Cycles in Cubic Graphs

100%
Bensmail J.
Discussiones Mathematicae Graph Theory
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2017
|
tom 37
|
nr 1
211-220
EN
In the context of a conjecture of Erdős and Gyárfás, we consider, for any q ≥ 2, the existence of q-power cycles (i.e., with length a power of q) in cubic graphs. We exhibit constructions showing that, for every q ≥ 3, there exist arbitrarily large cubic graphs with no q-power cycles. Concerning the remaining case q = 2 (which corresponds to the conjecture of Erdős and Gyárfás), we show that there exist arbitrarily large cubic graphs whose all 2-power cycles have length 4 only, or 8 only.
6
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A Fan-Type Heavy Pair Of Subgraphs For Pancyclicity Of 2-Connected Graphs

100%
Wideł W.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
|
nr 1
173-184
EN
Let G be a graph on n vertices and let H be a given graph. We say that G is pancyclic, if it contains cycles of all lengths from 3 up to n, and that it is H-f1-heavy, if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies [...] min⁡{dG(u),dG(v)}≥n+12 $\min \{ d_G (u),d_G (v)\} \ge {{n + 1} \over 2}$ . In this paper we prove that every 2-connected {K1,3, P5}-f1-heavy graph is pancyclic. This result completes the answer to the problem of finding f1-heavy pairs of subgraphs implying pancyclicity of 2-connected graphs.
7
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Dense Arbitrarily Partitionable Graphs

100%
Kalinowski R., Pilśniak M., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
|
nr 1
5-22
EN
A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with [...] ‖G‖ > (n−42)+12 $||G||\; > \;\left( {\matrix{{n - 4} \cr 2 \cr } } \right) + 12$ edges is AP or belongs to few classes of exceptional graphs.
8
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Matchings Extend to Hamiltonian Cycles in 5-Cube

100%
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.
9
Content available remote

Disjoint triangles and quadrilaterals in a graph

100%
Wang H.
Open Mathematics
|
2008
|
tom 6
|
nr 4
543-558
EN
Let n, s and t be three integers with s ≥ 1, t ≥ 0 and n = 3s + 4t. Let G be a graph of order n such that the minimum degree of G is at least (n + s)/2. Then G contains a 2-factor with s + t components such that s of them are triangles and t of them are quadrilaterals.
10
Content available remote

A Note on the Interval Function of a Disconnected Graph

76%
Changat M., Hossein Nezhad F., Mulder H. M., Narayanan N.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
39-48
EN
In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.
11
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The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

76%
Fernandes R.
Special Matrices
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2015
|
tom 3
|
nr 1
EN
The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial algorithm, cases of equality are computed for M2.
12
Content available remote

Rainbow Vertex-Connection and Forbidden Subgraphs

64%
Li W., Li X., Zhang J.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
143-154
EN
A path in a vertex-colored graph is called vertex-rainbow if its internal vertices have pairwise distinct colors. A vertex-colored graph G is rainbow vertex-connected if for any two distinct vertices of G, there is a vertex-rainbow path connecting them. For a connected graph G, the rainbow vertex-connection number of G, denoted by rvc(G), is defined as the minimum number of colors that are required to make G rainbow vertex-connected. In this paper, we find all the families ℱ of connected graphs with |ℱ| ∈ {1, 2}, for which there is a constant kℱ such that, for every connected ℱ-free graph G, rvc(G) ≤ diam(G) + kℱ, where diam(G) is the diameter of G.
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