scholarly journals Some sufficient conditions for a planar graph of maximum degree six to be Class 1

2006 ◽  
Vol 306 (13) ◽  
pp. 1440-1445 ◽  
Author(s):  
Yuehua Bu ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Maximum Degree ◽  
Sufficient Conditions ◽  
Class 1

scholarly journals Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Wenwen Zhang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Discharging Method ◽  
Class 1

In this paper, by applying the discharging method, we show that if G is a planar graph with a maximum degree of Δ = 6 that does not contain any adjacent 8-cycles, then G is of class 1.

APPROXIMATING THE NEAREST NEIGHBOR INTERCHARGE DISTANCE FOR NON-UNIFORM-DEGREE EVOLUTIONARY TREES

2001 ◽  
Vol 12 (04) ◽  
pp. 533-550 ◽  
Author(s):  
WING-KAI HON ◽  
TAK-WAH LAM
Keyword(s):  
Approximation Algorithms ◽  
Maximum Degree ◽  
Nearest Neighbor ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Approximation Ratio ◽  
Evolutionary Trees ◽  
Weighted Trees ◽  
Np Complete ◽  
Necessary And Sufficient

The nearest neighbor interchange (nni) distance is a classical metric for measuring the distance (dissimilarity) between evolutionary trees. It has been known that computing the nni distance is NP-complete. Existing approximation algorithms can attain an approximation ratio log n for unweighted trees and 4 log n for weighted trees; yet these algorithms are limited to degree-3 trees. This paper extends the study of nni distance to trees with non-uniform degrees. We formulate the necessary and sufficient conditions for nni transformation and devise more topology-sensitive approximation algorithms to handle trees with non-uniform degrees. The approximation ratios are respectively [Formula: see text] and [Formula: see text] for unweighted and weighted trees, where d ≥ 4 is the maximum degree of the input trees.

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

scholarly journals A sufficient condition for a planar graph to be class 1

2007 ◽  
Vol 385 (1-3) ◽  
pp. 71-77 ◽  
Author(s):  
Weifan Wang ◽  
Yongzhu Chen
Keyword(s):  
Planar Graph ◽  
Sufficient Condition ◽  
Class 1

A note on the 2-rainbow bondage numbers in graphs

2016 ◽  
Vol 09 (01) ◽  
pp. 1650013
Author(s):  
L. Asgharsharghi ◽  
S. M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Planar Graph ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Number Formula ◽  
Appl Math

A 2-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of [Formula: see text]. The rainbow bondage number [Formula: see text] of a graph [Formula: see text] with maximum degree at least two is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. Dehgardi, Sheikholeslami and Volkmann, [The [Formula: see text]-rainbow bondage number of a graph, Discrete Appl. Math. 174 (2014) 133–139] proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper, we generalize their result for graphs which admit a [Formula: see text]-cell embedding on a surface with non-negative Euler characteristic.

scholarly journals Near-Colorings: Non-Colorable Graphs and NP-Completeness

10.37236/3509 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. Montassier ◽  
P. Ochem
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Fixed Integer ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7. 

On Decay Centrality

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Nikolas Tsakas
Keyword(s):  
Numerical Simulations ◽  
Maximum Degree ◽  
High Probability ◽  
Sufficient Conditions ◽  
Optimal Choice ◽  
Simple Rule ◽  
Decay Parameter ◽  
Rule Of Thumb ◽  
Very High

AbstractWe establish a relationship between decay centrality and two widely used measures of centrality, namely degree and closeness. We show that for low values of the decay parameter the nodes with maximum decay centrality also have maximum degree, whereas for high values of the decay parameter they also maximize closeness. For intermediate values, we provide sufficient conditions that allow the comparison of decay centrality of different nodes and we show via numerical simulations that in the vast majority of networks, the nodes with maximum decay centrality are characterized by a threshold on the decay parameter below which they belong to the set of nodes with maximum degree and above which they belong to the set of nodes with maximum closeness. We also propose a simple rule of thumb that ensures a nearly optimal choice with very high probability.

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