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### On the stability for pancyclicity

Discussiones Mathematicae Graph Theory

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}\left(u\right)+{d}_{G}\left(v\right). 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...

### Problems remaining NP-complete for sparse or dense graphs

Discussiones Mathematicae Graph Theory

For each fixed pair α,c > 0 let INDEPENDENT SET ($m\le c{n}^{\alpha }$) and INDEPENDENT SET ($m\ge \left(ⁿ₂\right)-c{n}^{\alpha }$) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m\le c{n}^{\alpha }$ or $m\ge \left(ⁿ₂\right)-c{n}^{\alpha }$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ($m\le n+c{n}^{\alpha }$) and HAMILTONIAN PATH ($m\le n+c{n}^{\alpha }$) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m\le n+c{n}^{\alpha }$ 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...

### A new upper bound for the chromatic number of a graph

Discussiones Mathematicae Graph Theory

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 ${K}_{\omega +3}-C₅$.

### The cycle-complete graph Ramsey number r(C₅,K₇)

Discussiones Mathematicae Graph Theory

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.

### Bounds for the rainbow connection number of graphs

Discussiones Mathematicae Graph Theory

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.

### On the computational complexity of (O,P)-partition problems

Discussiones Mathematicae Graph Theory

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.

### The flower conjecture in special classes of graphs

Discussiones Mathematicae Graph Theory

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...

### Colouring graphs with prescribed induced cycle lengths

Discussiones Mathematicae Graph Theory

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 ${}^{I}\left(4,5\right)$, the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of ${}^{I}\left(4,5\right)$ are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind, i.e., we have...

### A lower bound on the independence number of a graph in terms of degrees

Discussiones Mathematicae Graph Theory

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.

### New sufficient conditions for hamiltonian and pancyclic graphs

Discussiones Mathematicae Graph Theory

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.

### Graphs with rainbow connection number two

Discussiones Mathematicae Graph Theory

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 $\left(\genfrac{}{}{0pt}{}{n-1}{2}\right)+1\le m\le \left(\genfrac{}{}{0pt}{}{n}{2}\right)-1$. We also characterize graphs with rainbow connection number two and large clique number.

### On Maximum Weight of a Bipartite Graph of Given Order and Size

Discussiones Mathematicae Graph Theory

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...

### Rainbow Connection Number of Dense Graphs

Discussiones Mathematicae Graph Theory

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.

### The Ramsey number r(C₇,C₇,C₇)

Discussiones Mathematicae Graph Theory

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.

### Closure for spanning trees and distant area

Discussiones Mathematicae Graph Theory

A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with $de{g}_{G}u+de{g}_{G}v\ge n-1$, G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on $de{g}_{G}u+de{g}_{G}v$ and the structure of the distant area for u and v. We prove that if the...

### Rainbow Connection In Sparse Graphs

Discussiones Mathematicae Graph Theory

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.

### Extending the MAX Algorithm for Maximum Independent Set

Discussiones Mathematicae Graph Theory

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.

### Graphs with 3-Rainbow Index n − 1 and n − 2

Discussiones Mathematicae Graph Theory

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...

### Graphs with 4-Rainbow Index 3 and n − 1

Discussiones Mathematicae Graph Theory

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)....

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