### Abstract

Suppose G is a graph on n vertices with minimum degree r. Using standard random methods it is shown that there exists a two-coloring of the vertices of G with colors, +1 and -1, such that all closed neighborhoods contain more 1's than -1's, and all together the number of 1's does not exceed the number of -1's by more than (4logr/r+1/r)n. For large r this greatly improves earlier results and is almost optimal, since starting with an Hadamard matrix of order r, a bipartite r-regular graph is constructed on 4r vertices with signed domination number at least (1/2) r-O(1). The determination of lim_{n→∞}γ_{s}(G)/n remains open and is conjectured to be Θ(1/r).

Original language | English |
---|---|

Pages (from-to) | 223-239 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 76 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 1999 |

### Fingerprint

### Keywords

- Discrepancy
- Domination
- Hadamard matrices
- Random covering of graphs and hypergraphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*76*(2), 223-239. https://doi.org/10.1006/jctb.1999.1905

**Signed Domination in Regular Graphs and Set-Systems.** / Füredi, Z.; Mubayi, Dhruv.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 76, no. 2, pp. 223-239. https://doi.org/10.1006/jctb.1999.1905

}

TY - JOUR

T1 - Signed Domination in Regular Graphs and Set-Systems

AU - Füredi, Z.

AU - Mubayi, Dhruv

PY - 1999/7

Y1 - 1999/7

N2 - Suppose G is a graph on n vertices with minimum degree r. Using standard random methods it is shown that there exists a two-coloring of the vertices of G with colors, +1 and -1, such that all closed neighborhoods contain more 1's than -1's, and all together the number of 1's does not exceed the number of -1's by more than (4logr/r+1/r)n. For large r this greatly improves earlier results and is almost optimal, since starting with an Hadamard matrix of order r, a bipartite r-regular graph is constructed on 4r vertices with signed domination number at least (1/2) r-O(1). The determination of limn→∞γs(G)/n remains open and is conjectured to be Θ(1/r).

AB - Suppose G is a graph on n vertices with minimum degree r. Using standard random methods it is shown that there exists a two-coloring of the vertices of G with colors, +1 and -1, such that all closed neighborhoods contain more 1's than -1's, and all together the number of 1's does not exceed the number of -1's by more than (4logr/r+1/r)n. For large r this greatly improves earlier results and is almost optimal, since starting with an Hadamard matrix of order r, a bipartite r-regular graph is constructed on 4r vertices with signed domination number at least (1/2) r-O(1). The determination of limn→∞γs(G)/n remains open and is conjectured to be Θ(1/r).

KW - Discrepancy

KW - Domination

KW - Hadamard matrices

KW - Random covering of graphs and hypergraphs

UR - http://www.scopus.com/inward/record.url?scp=0242364590&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242364590&partnerID=8YFLogxK

U2 - 10.1006/jctb.1999.1905

DO - 10.1006/jctb.1999.1905

M3 - Article

AN - SCOPUS:0242364590

VL - 76

SP - 223

EP - 239

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -