A note on the independent domination number of subset graph

Xue-Gang Chen; De-xiang Ma; Hua Ming Xing; Liang Sun

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 2, page 511-517
  • ISSN: 0011-4642

Abstract

top
The independent domination number i ( G ) (independent number β ( G ) ) is the minimum (maximum) cardinality among all maximal independent sets of G . Haviland (1995) conjectured that any connected regular graph G of order n and degree δ ≤ 1 2 n satisfies i ( G ) ≤ ⌈ 2 n 3 δ ⌉ 1 2 δ . For 1 ≤ k ≤ l ≤ m , the subset graph S m ( k , l ) is the bipartite graph whose vertices are the k - and l -subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i ( S m ( k , l ) ) and prove that if k + l = m then Haviland’s conjecture holds for the subset graph S m ( k , l ) . Furthermore, we give the exact value of β ( S m ( k , l ) ) .

How to cite

top

Chen, Xue-Gang, et al. "A note on the independent domination number of subset graph." Czechoslovak Mathematical Journal 55.2 (2005): 511-517. <http://eudml.org/doc/30965>.

@article{Chen2005,
abstract = {The independent domination number $i(G)$ (independent number $\beta (G)$) is the minimum (maximum) cardinality among all maximal independent sets of $G$. Haviland (1995) conjectured that any connected regular graph $G$ of order $n$ and degree $\delta \le \frac\{1\}\{2\}\{n\}$ satisfies $i(G)\le \lceil \frac\{2n\}\{3\delta \}\rceil \frac\{1\}\{2\}\{\delta \}$. For $1\le k\le l\le m$, the subset graph $S_\{m\}(k,l)$ is the bipartite graph whose vertices are the $k$- and $l$-subsets of an $m$ element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for $i(S_\{m\}(k,l))$ and prove that if $k+l=m$ then Haviland’s conjecture holds for the subset graph $S_\{m\}(k,l)$. Furthermore, we give the exact value of $\beta (S_\{m\}(k,l))$.},
author = {Chen, Xue-Gang, Ma, De-xiang, Xing, Hua Ming, Sun, Liang},
journal = {Czechoslovak Mathematical Journal},
keywords = {independent domination number; independent number; subset graph; independent domination number; independent number; subset graph},
language = {eng},
number = {2},
pages = {511-517},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the independent domination number of subset graph},
url = {http://eudml.org/doc/30965},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Chen, Xue-Gang
AU - Ma, De-xiang
AU - Xing, Hua Ming
AU - Sun, Liang
TI - A note on the independent domination number of subset graph
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 2
SP - 511
EP - 517
AB - The independent domination number $i(G)$ (independent number $\beta (G)$) is the minimum (maximum) cardinality among all maximal independent sets of $G$. Haviland (1995) conjectured that any connected regular graph $G$ of order $n$ and degree $\delta \le \frac{1}{2}{n}$ satisfies $i(G)\le \lceil \frac{2n}{3\delta }\rceil \frac{1}{2}{\delta }$. For $1\le k\le l\le m$, the subset graph $S_{m}(k,l)$ is the bipartite graph whose vertices are the $k$- and $l$-subsets of an $m$ element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for $i(S_{m}(k,l))$ and prove that if $k+l=m$ then Haviland’s conjecture holds for the subset graph $S_{m}(k,l)$. Furthermore, we give the exact value of $\beta (S_{m}(k,l))$.
LA - eng
KW - independent domination number; independent number; subset graph; independent domination number; independent number; subset graph
UR - http://eudml.org/doc/30965
ER -

References

top
  1. Independence graphs, Proc. 5th Southeast Conf. Comb. Graph Theor. Comput, Utilitas Math., Boca Raton, 1974, pp. 471–491. (1974) MR0357174
  2. 10.1016/0012-365X(88)90076-3, Discrete Math. 70 (1988), 17–20. (1988) MR0943719DOI10.1016/0012-365X(88)90076-3
  3. 10.1016/0012-365X(91)90318-V, Discrete Math. 94 (1991), 95–101. (1991) Zbl0758.05061MR1139586DOI10.1016/0012-365X(91)90318-V
  4. 10.1016/0012-365X(94)00022-B, Discrete Math. 143 (1995), 275–280. (1995) Zbl0838.05065MR1344759DOI10.1016/0012-365X(94)00022-B
  5. 10.1016/0012-365X(96)00025-8, Discrete Math. 158 (1996), 87–98. (1996) MR1411112DOI10.1016/0012-365X(96)00025-8
  6. 10.1016/0012-365X(81)90268-5, Discrete Math. 33 (1981), 249–258. (1981) MR0602041DOI10.1016/0012-365X(81)90268-5
  7. 10.1016/S0012-365X(98)00350-1, Discrete Math. 202 (1999), 135–144. (1999) MR1694509DOI10.1016/S0012-365X(98)00350-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.