A new upper bound for the independence number of edge chromatic critical graphs

2010 ◽  
Vol 68 (3) ◽  
pp. 202-212 ◽  
Author(s):  
Rong Luo ◽  
Yue Zhao
Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.


10.37236/7272 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Xujun Liu

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and  let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás proved the upper bound  $i(3,\infty) < \frac{6}{13}$  in 1981, and McKay improved it to $i(3,\infty) < 0.45537$in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \leqslant 0.454.$


10.37236/2598 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Tom Bohman ◽  
Ron Holzman ◽  
Venkatesh Natarajan

We give an upper bound on the independence number of the cube of the odd cycle $C_{8n+5}$. The best known lower bound is conjectured to be the truth; we prove the conjecture in the case $8n+5$ prime and, within $2$, for general $n$.


2018 ◽  
Vol 13 (01) ◽  
pp. 2050013
Author(s):  
H. S. Mehta ◽  
U. P. Acharya

The tensor product and the Cartesian product of two graphs are very well-known graph products and studied in detail. Many graph parameters, particularly independence number, have been studied for these graph products. These two graph products have been generalized by [Formula: see text]-tensor product and [Formula: see text]-Cartesian product, respectively, and studied in detail. In this paper, we discuss the independence number for [Formula: see text]-tensor product [Formula: see text] and [Formula: see text]-Cartesian product [Formula: see text]. In general, we obtain lower bound and upper bound for the independence number.


2012 ◽  
Vol 84 (1-2) ◽  
pp. 71-76 ◽  
Author(s):  
Sizhong Zhou ◽  
Lan Xu ◽  
Zhiren Sun

2000 ◽  
Vol 32 (2) ◽  
pp. 137-140 ◽  
Author(s):  
Gunnar Brinkmann ◽  
Sheshayya A. Choudum ◽  
Stefan Grünewald ◽  
Eckhard Steffen

2014 ◽  
Vol 34 (3) ◽  
pp. 577 ◽  
Author(s):  
Lianying Miao ◽  
Zhengke Miao ◽  
Shiyou Pang ◽  
Wenyao Song

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