scholarly journals Independence Number and Disjoint Theta Graphs

10.37236/637 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Shinya Fujita ◽  
Colton Magnant
Keyword(s):  
Independence Number ◽  
Disjoint Paths ◽  
Ramsey Problem ◽  
Fixed Integer ◽  
Sharp Bounds ◽  
Vertex Disjoint ◽  
The Relationship

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a $\theta$-graph to be a pair of vertices $u, v$ with three internally disjoint paths joining $u$ to $v$. Given an independence number $\alpha$ and a fixed integer $k$, the results contained in this paper provide sharp bounds on the order $f(k, \alpha)$ of a graph with independence number $\alpha(G) \leq \alpha$ which contains no $k$ disjoint $\theta$-graphs. Since every $\theta$-graph contains an even cycle, these results provide $k$ disjoint even cycles in graphs of order at least $f(k, \alpha) + 1$. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

The Hyper-Zagreb Index and Some Properties of Graphs

2020 ◽  
pp. 120-134 ◽  
Author(s):  
Rao Li
Keyword(s):  
Topological Index ◽  
Sufficient Conditions ◽  
Disjoint Paths ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Second Zagreb Index ◽  
Vertex Disjoint ◽  
Hamiltonian Properties

Let G = (V(G), E(G)) be a graph. The complement of G is denoted by Gc. The forgotten topological index of G, denoted F(G), is defined as the sum of the cubes of the degrees of all the vertices in G. The second Zagreb index of G, denoted M2(G), is defined as the sum of the products of the degrees of pairs of adjacent vertices in G. A graph Gisk-Hamiltonian if for all X ⊂V(G) with|X| ≤ k, the subgraph induced byV(G) - Xis Hamiltonian. Clearly, G is 0-Hamiltonian if and only if G is Hamiltonian. A graph Gisk-path-coverableifV(G) can be covered bykor fewer vertex-disjoint paths. Using F(Gc) and M2(Gc), Li obtained several sufficient conditions for Hamiltonian and traceable graphs (Rao Li, Topological Indexes and Some Hamiltonian Properties of Graphs). In this chapter, the author presents sufficient conditions based upon F(Gc) and M2(Gc)for k-Hamiltonian, k-edge-Hamiltonian, k-path-coverable, k-connected, and k-edge-connected graphs.

scholarly journals Vertex-Disjoint Paths in a 3-Ary n-Cube with Faulty Vertices

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Xiaolei Ma ◽  
Shiying Wang
Keyword(s):  
Interconnection Networks ◽  
Transmission Rate ◽  
Research Topic ◽  
Disjoint Paths ◽  
Important Research ◽  
Important Research Topic ◽  
Vertex Disjoint

The construction of vertex-disjoint paths (disjoint paths) is an important research topic in various kinds of interconnection networks, which can improve the transmission rate and reliability. The k-ary n-cube is a family of popular networks. In this paper, we determine that there are m2≤m≤n disjoint paths in 3-ary n-cube covering Qn3−F from S to T (many-to-many) with F≤2n−2m and from s to T (one-to-many) with F≤2n−m−1 where s is in a fault-free cycle of length three.

Gossiping in vertex-disjoint paths mode in d-dimensional grids and planar graphs

1993 ◽  
pp. 200-211 ◽  
Author(s):  
Juraj Hromkovič ◽  
Ralf Klasing ◽  
Elena A. Stöhr ◽  
Hubert Wagener
Keyword(s):  
Planar Graphs ◽  
Disjoint Paths ◽  
Vertex Disjoint

Turán numbers of theta graphs

2020 ◽  
Vol 29 (4) ◽  
pp. 495-507 ◽  
Author(s):  
Boris Bukh ◽  
Michael Tait
Keyword(s):  
Disjoint Paths ◽  
Vertex Disjoint

AbstractThe theta graph ${\Theta _{\ell ,t}}$ consists of two vertices joined by t vertex-disjoint paths, each of length $\ell $ . For fixed odd $\ell $ and large t, we show that the largest graph not containing ${\Theta _{\ell ,t}}$ has at most ${c_\ell }{t^{1 - 1/\ell }}{n^{1 + 1/\ell }}$ edges and that this is tight apart from the value of ${c_\ell }$ .

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