New sharp lower bound for the quorum coloring number of trees
•The alliance partition number of a tree has a new sharp lower bound computable in linear time.•This new lower bound is better than all those that have been previously established.•There is a relationship between the order, the diameter, the vertices degree and the matching number of a subgraph of a...
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| Published in: | Information processing letters Vol. 178; p. 106297 |
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| Format: | Journal Article |
| Language: | English |
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Elsevier B.V
01.11.2022
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | •The alliance partition number of a tree has a new sharp lower bound computable in linear time.•This new lower bound is better than all those that have been previously established.•There is a relationship between the order, the diameter, the vertices degree and the matching number of a subgraph of a tree.•The alliance partition number of a binary tree is computable in linear time.
A partition π={V1,V2,...,Vk} of the vertex set V of a graph G into k color classes Vi, with 1≤i≤k is called a quorum coloring if for every vertex v∈V, at least half of the vertices in the closed neighborhood N[v] of v have the same color as v. The maximum cardinality of a quorum coloring of G is called the quorum coloring number of G and is denoted by ψq(G). A quorum coloring of order ψq(G) is a ψq-coloring. In this paper, we partially answer an open problem concerning quorum colorings of graphs. Namely, we improve a sharp lower bound given in 2012 by Eroh and Gera on the quorum coloring number of a nontrivial tree, and show that our new lower bound can be computed in linear time. Moreover, we show that this bound is attained by all non trivial binary trees. |
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| AbstractList | •The alliance partition number of a tree has a new sharp lower bound computable in linear time.•This new lower bound is better than all those that have been previously established.•There is a relationship between the order, the diameter, the vertices degree and the matching number of a subgraph of a tree.•The alliance partition number of a binary tree is computable in linear time.
A partition π={V1,V2,...,Vk} of the vertex set V of a graph G into k color classes Vi, with 1≤i≤k is called a quorum coloring if for every vertex v∈V, at least half of the vertices in the closed neighborhood N[v] of v have the same color as v. The maximum cardinality of a quorum coloring of G is called the quorum coloring number of G and is denoted by ψq(G). A quorum coloring of order ψq(G) is a ψq-coloring. In this paper, we partially answer an open problem concerning quorum colorings of graphs. Namely, we improve a sharp lower bound given in 2012 by Eroh and Gera on the quorum coloring number of a nontrivial tree, and show that our new lower bound can be computed in linear time. Moreover, we show that this bound is attained by all non trivial binary trees. |
| ArticleNumber | 106297 |
| Author | Sahbi, Rafik |
| Author_xml | – sequence: 1 givenname: Rafik orcidid: 0000-0003-1175-5137 surname: Sahbi fullname: Sahbi, Rafik email: r.sahbi@g.essa-alger.edu.dz organization: Department of the Preparatory Training, Algiers Higher School of Applied Sciences, B.P. 474, Martyrs Square, Algiers 16001, Algeria |
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| Cites_doi | 10.1016/j.akcej.2017.05.002 10.1016/j.akcej.2019.12.010 10.1016/0020-0190(80)90140-4 10.5614/ejgta.2014.2.1.7 10.1051/ro/2021116 10.1016/j.dam.2018.03.060 |
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| Keywords | Matchings Algorithms Defensive alliances Linear-time algorithms Binary trees |
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| References | Savage (br0120) 1980; 10 R. Sahbi, Y. Belkina, A. Bennadji, Sub-quorum colorings of some infinite families of caterpillars, submitted for publication. Sahbi (br0090) 2020; 17 Sahbi, Chellali (br0110) 2018; 247 Shafique (br0130) 2004 Belkina, Bennadji (br0140) 2021 Hedetniemi, Hedetniemi, Laskar, Mulder (br0070) 2013; 10 Haynes, Lachniet (br0040) 2007; 4 Fricke, Lawson, Haynes, Hedetniemi, Hedetniemi (br0030) 2003; 38 Sahbi (br0100) 2021; 55 Hedetniemi, Hedetniemi, Kristiansen (br0050) 2004; 48 Fernaua, Rodriguez-Velazquez (br0020) 2014; 2 Eroh, Gera (br0010) 2012; 103 Ouazine, Slimani, Tari (br0080) 2018; 15 Ouazine (10.1016/j.ipl.2022.106297_br0080) 2018; 15 Savage (10.1016/j.ipl.2022.106297_br0120) 1980; 10 Eroh (10.1016/j.ipl.2022.106297_br0010) 2012; 103 Fernaua (10.1016/j.ipl.2022.106297_br0020) 2014; 2 Hedetniemi (10.1016/j.ipl.2022.106297_br0070) 2013; 10 Sahbi (10.1016/j.ipl.2022.106297_br0110) 2018; 247 Hedetniemi (10.1016/j.ipl.2022.106297_br0050) 2004; 48 Belkina (10.1016/j.ipl.2022.106297_br0140) 2021 Fricke (10.1016/j.ipl.2022.106297_br0030) 2003; 38 Sahbi (10.1016/j.ipl.2022.106297_br0100) 2021; 55 10.1016/j.ipl.2022.106297_br0150 Shafique (10.1016/j.ipl.2022.106297_br0130) 2004 Sahbi (10.1016/j.ipl.2022.106297_br0090) 2020; 17 Haynes (10.1016/j.ipl.2022.106297_br0040) 2007; 4 |
| References_xml | – volume: 103 start-page: 519 year: 2012 end-page: 529 ident: br0010 article-title: Alliance partition number in graphs publication-title: Ars Comb. – volume: 10 start-page: 202 year: 1980 end-page: 205 ident: br0120 article-title: Maximum matchings and trees publication-title: Inf. Process. Lett. – volume: 247 start-page: 294 year: 2018 end-page: 299 ident: br0110 article-title: On some open problems concerning quorum colorings of graphs publication-title: Discrete Appl. Math. – volume: 10 start-page: 97 year: 2013 end-page: 109 ident: br0070 article-title: Quorum colorings of graphs publication-title: AKCE Int. J. Graphs Comb. – year: 2021 ident: br0140 article-title: Sous-quorum-colorations de quelques classes d'arbres – year: 2004 ident: br0130 article-title: Partitioning a graph in alliances and its application to data clustering – volume: 4 start-page: 51 year: 2007 end-page: 59 ident: br0040 article-title: The alliance partition number of grid graphs publication-title: AKCE Int. J. Graphs Comb. – volume: 38 start-page: 37 year: 2003 end-page: 41 ident: br0030 article-title: A note on defensive alliances in graphs publication-title: Bull. Inst. Comb. Appl. – volume: 48 start-page: 157 year: 2004 end-page: 177 ident: br0050 article-title: Alliances in graphs publication-title: J. Comb. Math. Comb. Comput. – volume: 17 start-page: 784 year: 2020 end-page: 787 ident: br0090 article-title: On the complexity of some quorum colorings problems of graphs publication-title: AKCE Int. J. Graphs Comb. – volume: 15 start-page: 115 year: 2018 end-page: 154 ident: br0080 article-title: Alliances in graphs: parameters, properties and applications-a survey publication-title: AKCE Int. J. Graphs Comb. – volume: 2 start-page: 70 year: 2014 end-page: 86 ident: br0020 article-title: A survey on alliances and related parameters in graphs publication-title: Electron. J. Graph Theory Appl. – reference: R. Sahbi, Y. Belkina, A. Bennadji, Sub-quorum colorings of some infinite families of caterpillars, submitted for publication. – volume: 55 start-page: 2385 year: 2021 end-page: 2394 ident: br0100 article-title: Solutions to four open problems on quorum colorings of graphs publication-title: RAIRO Oper. Res. – volume: 38 start-page: 37 year: 2003 ident: 10.1016/j.ipl.2022.106297_br0030 article-title: A note on defensive alliances in graphs publication-title: Bull. Inst. Comb. Appl. – volume: 15 start-page: 115 year: 2018 ident: 10.1016/j.ipl.2022.106297_br0080 article-title: Alliances in graphs: parameters, properties and applications-a survey publication-title: AKCE Int. J. Graphs Comb. doi: 10.1016/j.akcej.2017.05.002 – year: 2021 ident: 10.1016/j.ipl.2022.106297_br0140 – volume: 17 start-page: 784 issue: 3 year: 2020 ident: 10.1016/j.ipl.2022.106297_br0090 article-title: On the complexity of some quorum colorings problems of graphs publication-title: AKCE Int. J. Graphs Comb. doi: 10.1016/j.akcej.2019.12.010 – volume: 4 start-page: 51 issue: 1 year: 2007 ident: 10.1016/j.ipl.2022.106297_br0040 article-title: The alliance partition number of grid graphs publication-title: AKCE Int. J. Graphs Comb. – volume: 10 start-page: 202 issue: 4/5 year: 1980 ident: 10.1016/j.ipl.2022.106297_br0120 article-title: Maximum matchings and trees publication-title: Inf. Process. Lett. doi: 10.1016/0020-0190(80)90140-4 – volume: 10 start-page: 97 issue: 1 year: 2013 ident: 10.1016/j.ipl.2022.106297_br0070 article-title: Quorum colorings of graphs publication-title: AKCE Int. J. Graphs Comb. – year: 2004 ident: 10.1016/j.ipl.2022.106297_br0130 – volume: 103 start-page: 519 year: 2012 ident: 10.1016/j.ipl.2022.106297_br0010 article-title: Alliance partition number in graphs publication-title: Ars Comb. – volume: 2 start-page: 70 issue: 1 year: 2014 ident: 10.1016/j.ipl.2022.106297_br0020 article-title: A survey on alliances and related parameters in graphs publication-title: Electron. J. Graph Theory Appl. doi: 10.5614/ejgta.2014.2.1.7 – volume: 55 start-page: 2385 issue: 4 year: 2021 ident: 10.1016/j.ipl.2022.106297_br0100 article-title: Solutions to four open problems on quorum colorings of graphs publication-title: RAIRO Oper. Res. doi: 10.1051/ro/2021116 – volume: 48 start-page: 157 year: 2004 ident: 10.1016/j.ipl.2022.106297_br0050 article-title: Alliances in graphs publication-title: J. Comb. Math. Comb. Comput. – ident: 10.1016/j.ipl.2022.106297_br0150 – volume: 247 start-page: 294 year: 2018 ident: 10.1016/j.ipl.2022.106297_br0110 article-title: On some open problems concerning quorum colorings of graphs publication-title: Discrete Appl. Math. doi: 10.1016/j.dam.2018.03.060 |
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| Snippet | •The alliance partition number of a tree has a new sharp lower bound computable in linear time.•This new lower bound is better than all those that have been... |
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| SubjectTerms | Algorithms Binary trees Defensive alliances Linear-time algorithms Matchings |
| Title | New sharp lower bound for the quorum coloring number of trees |
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