Randomly colouring graphs with lower bounds on girth and maximum degree

2001 ◽  
Author(s):  
M. Dyer ◽  
A. Frieze
Keyword(s):  
Lower Bounds ◽  
Maximum Degree

Randomly coloring graphs with lower bounds on girth and maximum degree

2003 ◽  
Vol 23 (2) ◽  
pp. 167-179 ◽  
Author(s):  
Martin Dyer ◽  
Alan Frieze
Keyword(s):  
Lower Bounds ◽  
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 Tight lower bounds on the matching number in a graph with given maximum degree

2018 ◽  
Vol 89 (2) ◽  
pp. 115-149 ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Lower Bounds ◽  
Maximum Degree ◽  
Matching Number

scholarly journals New Bounds for Topological Indices on Trees through Generalized Methods

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1097 ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Chemical Compounds ◽  
Wiener Index ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Unified Approach ◽  
Wide Family ◽  
New Inequalities

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index.

Girth and Independence Ratio

1982 ◽  
Vol 25 (2) ◽  
pp. 179-186 ◽  
Author(s):  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bounds ◽  
Maximum Degree ◽  
Cubic Graphs

AbstractLower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.

scholarly journals Difference betwwen the maximum degree and the index of a graph

2019 ◽  
Vol 54 (4) ◽  
pp. 434-440
Author(s):  
V. I. Benediktovich
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Lower Boundary ◽  
Arbitrary Graph ◽  
Analogous Statement ◽  
Graph Classes ◽  
The Difference ◽  
A Chain

An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.

scholarly journals Colouring the Square of the Cartesian Product of Trees

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
International Audience

Graphs and Algorithms International audience We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.

scholarly journals The treewidth of 2-section of hypergraphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Ke Liu ◽  
Mei Lu
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper And Lower Bounds ◽  
Average Rank ◽  
Linear Hypergraph ◽  
Algorithmic Graph Theory

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

scholarly journals Maximum Induced Forests in Graphs of Bounded Treewidth

10.37236/3826 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Glenn G. Chappell ◽  
Michael J. Pelsmajer
Keyword(s):  
Lower Bounds ◽  
Nonnegative Integer ◽  
Maximum Degree ◽  
Open Problems ◽  
Maximum Order ◽  
Bounded Treewidth

Given a nonnegative integer $d$ and a graph $G$, let $f_d(G)$ be the maximum order of an induced forest in $G$ having maximum degree at most $d$. We seek lower bounds for $f_d(G)$ based on the order and treewidth of $G$.We show that, for all $k,d\ge 2$ and $n\ge 1$, if $G$ is a graph with order $n$ and treewidth at most $k$, then $f_d(G)\ge\lceil{(2dn+2)/(kd+d+1)}\rceil$, unless $G\in\{K_{1,1,3},K_{2,3}\}$ and $k=d=2$. We give examples that show that this bound is tight to within $1$.We conjecture a bound for $d=1$: $f_1(G) \ge\lceil{2n/(k+2)}\rceil$, which would also be tight to within $1$, and we prove it for $k\le 3$. For $k\ge 4$ the conjecture remains open, and we prove a weaker bound: $f_1(G)\ge (2n+2)/(2k+3)$. We also examine the cases $d=0$ and $k=0,1$.Lastly, we consider open problems relating to $f_d$ for graphs on a given surface, rather than graphs of bounded treewidth.

scholarly journals New Approach to the $k$-Independence Number of a Graph

10.37236/2646 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Yair Caro ◽  
Adriana Hansberg
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Independence Number ◽  
Independent Set ◽  
Average Degree ◽  
Maximum Cardinality ◽  
New Approach

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of a $k$-independent set of $G$. We prove that, for a graph $G$ on $n$ vertices and average degree $d$, $\alpha_k(G) \ge \frac{k+1}{\lceil d \rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on $k$-independence, J. Graph Theory 15 (1991), 99-107].

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