MaxDDBS Problem on Beneš Network

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741003
Author(s):  
NOVI H. BONG ◽  
JOE RYAN ◽  
KIKI A. SUGENG
Keyword(s):  
Lower Bound ◽  
Maximum Degree ◽  
Benes Network ◽  
Host Graph

Maximum degree-diameter bounded subgraph problem is a problem of constructing the largest possible subgraph of given degree and diameter in a graph. This problem can be considered as a degree-diameter problem restricted to certain host graphs. The MaxDDBS problem with Beneš network as the host graph is discussed in this paper. Beneš network contains a back-to-back buttery network. Even though both networks have maximum degree 4, the structure of their maximum subgraphs are different. We give the constructive lower bound of the largest subgraph of Beneš network of various maximum degrees.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

BIPARTITE SUBGRAPHS OF -FREE GRAPHS

2017 ◽  
Vol 96 (1) ◽  
pp. 1-13 ◽  
Author(s):  
QINGHOU ZENG ◽  
JIANFENG HOU
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Bipartite Subgraph

For a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. For an integer $m$ and for a fixed graph $H$, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$ as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. We give a general lower bound for $f(m,H)$ which extends a result of Erdős and Lovász and we study this function for any bipartite graph $H$ with maximum degree at most $t\geq 2$ on one side.

scholarly journals A note on domino treewidth

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Hans L. Bodlaender
Keyword(s):  
Lower Bound ◽  
Simple Proof ◽  
Maximum Degree ◽  
International Audience

International audience In [DO95], Ding and Oporowski proved that for every k, and d, there exists a constant c_k,d, such that every graph with treewidth at most k and maximum degree at most d has domino treewidth at most c_k,d. This note gives a new simple proof of this fact, with a better bound for c_k,d, namely (9k+7)d(d+1) -1. It is also shown that a lower bound of Ω (kd) holds: there are graphs with domino treewidth at least 1/12 × kd-1, treewidth at most k, and maximum degree at most d, for many values k and d. The domino treewidth of a tree is at most its maximum degree.

scholarly journals Remarks on upper and lower bounds formatching sequencibility of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2091-2099
Author(s):  
Shuya Chiba ◽  
Yuji Nakano
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Hamilton Cycle ◽  
Upper And Lower Bounds ◽  
Linear Ordering ◽  
The Relationship

In 2008, Alspach [The Wonderful Walecki Construction, Bull. Inst. Combin. Appl. 52 (2008) 7-20] defined the matching sequencibility of a graph G to be the largest integer k such that there exists a linear ordering of its edges so that every k consecutive edges in the linear ordering form a matching of G, which is denoted by ms(G). In this paper, we show that every graph G of size q and maximum degree ? satisfies 1/2?q/?+1? ? ms(G) ? ?q?1/??1? by using the edge-coloring of G, and we also improve this lower bound for some particular graphs. We further discuss the relationship between the matching sequencibility and a conjecture of Seymour about the existence of the kth power of a Hamilton cycle.

scholarly journals Restrained Italian domination in graphs

2021 ◽  
Vol 55 (2) ◽  
pp. 319-332
Author(s):  
Babak Samadi ◽  
Morteza Alishahi ◽  
Iman Masoumi ◽  
Doost Ali Mojdeh
Keyword(s):  
Lower Bound ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Double Star ◽  
Chordal Graphs ◽  
Domination In Graphs ◽  
General Graphs ◽  
Dominating Function

For a graph G = (V(G), E(G)), an Italian dominating function (ID function) f : V(G) → {0,1,2} has the property that for every vertex v ∈ V(G) with f(v) = 0, either v is adjacent to a vertex assigned 2 under f or v is adjacent to least two vertices assigned 1 under f. The weight of an ID function is ∑v∈V(G) f(v). The Italian domination number is the minimum weight taken over all ID functions of G. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) f of G is an ID function of G for which the subgraph induced by {v ∈ V(G) | f(v) = 0} has no isolated vertices, and the restrained Italian domination number γrI (G) is the minimum weight taken over all RID functions of G. We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that γrI(T) for a tree T of order n ≥ 3 different from the double star S2,2 can be bounded from below by (n + 3)/2. Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphs G with small or large γrI (G).

scholarly journals Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

scholarly journals A New Lower Bound on the Independence Number of a Graph and Applications

10.37236/3601 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Christian Löwenstein ◽  
Justin Southey ◽  
Anders Yeo
Keyword(s):  
Lower Bound ◽  
Maximum Degree ◽  
Independence Number ◽  
Maximum Clique ◽  
Independent Set ◽  
Uniform Hypergraph ◽  
Maximum Cardinality ◽  
Uniform Hypergraphs ◽  
Transversal Number ◽  
Graph Parameters

The independence number of a graph $G$, denoted $\alpha(G)$, is the maximum cardinality of an independent set of vertices in $G$. The independence number is one of the most fundamental and well-studied graph parameters. In this paper, we strengthen a result of Fajtlowicz [Combinatorica 4 (1984), 35-38] on the independence of a graph given its maximum degree and maximum clique size. As a consequence of our result we give bounds on the independence number and transversal number of $6$-uniform hypergraphs with maximum degree three. This gives support for a conjecture due to Tuza and Vestergaard [Discussiones Math. Graph Theory 22 (2002), 199-210] that if $H$ is a $3$-regular $6$-uniform hypergraph of order $n$, then $\tau(H) \le n/4$.

A Linear Time Algorithm for Embedding Christmas Trees into Certain Trees

2015 ◽  
Vol 25 (04) ◽  
pp. 1550008
Author(s):  
Indra Rajasingh ◽  
R. Sundara Rajan ◽  
Paul Manuel
Keyword(s):  
Lower Bound ◽  
Interconnection Network ◽  
Linear Time ◽  
Graph Embedding ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Christmas Trees ◽  
Host Graph

Graph embedding is an important technique that maps a logical graph into a host graph, usually an interconnection network. In this paper, we compute the exact wirelength of embedding Christmas trees into trees. Moreover, we present an algorithm for embedding Christmas trees into caterpillars with dilation 3 proving that the lower bound obtained in [30] is sharp. Further, we solve the maximum subgraph problem for Christmas trees and provide a linear time algorithm to compute the exact wirelength of embedding Christmas trees into trees.

scholarly journals On the Chromatic Number of Matching Kneser Graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 1-21
Author(s):  
Meysam Alishahi ◽  
Hajiabolhassan Hossein
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Large Family ◽  
Permutation Graphs ◽  
Kneser Graph ◽  
Ulam Theorem ◽  
Turán Number ◽  
Vertex Set ◽  
Kneser Graphs ◽  
Host Graph

AbstractIn an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

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