Pub Date : 2020-07-30DOI: 10.11575/CDM.V15I2.62352
Oguz Dogan
In this study, we show that necessary conditions for $Q_4$-factorization of $lambda{K_n}$ and $lambda{K_{x(m)}}$ (complete $x$ partite graph with parts of size $m$) are sufficient. We proved that there exists a $Q_4$-factorization of $lambda{K_{x(m)}}$ if and only if $mxequiv{0} pmod{16}$ and $lambda{m(x-1)}equiv{0}pmod{4}$. This result immediately gives that $lambda K_n$ has a $Q_4$-factorization if and only if $nequiv 0 pmod{16}$ and $lambda equiv 0 pmod{4}$.
{"title":"$Q_4$-Factorization of $lambda K_n$ and $lambda K_x(m)$","authors":"Oguz Dogan","doi":"10.11575/CDM.V15I2.62352","DOIUrl":"https://doi.org/10.11575/CDM.V15I2.62352","url":null,"abstract":"In this study, we show that necessary conditions for $Q_4$-factorization of $lambda{K_n}$ and $lambda{K_{x(m)}}$ (complete $x$ partite graph with parts of size $m$) are sufficient. We proved that there exists a $Q_4$-factorization of $lambda{K_{x(m)}}$ if and only if $mxequiv{0} pmod{16}$ and $lambda{m(x-1)}equiv{0}pmod{4}$. This result immediately gives that $lambda K_n$ has a $Q_4$-factorization if and only if $nequiv 0 pmod{16}$ and $lambda equiv 0 pmod{4}$.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"15 1","pages":"18-26"},"PeriodicalIF":0.5,"publicationDate":"2020-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45695220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-11DOI: 10.11575/CDM.V15I1.62808
R. Dawson
We show that there is a unique way to partition a $5times 5$ array of lattice points into restrictions of five circles. This result is extended to the $6times 5$ array, and used to show the optimality of a six-circle solution for the $6times 6$ array.
{"title":"Partitioning the $5times 5$ array into restrictions of circles","authors":"R. Dawson","doi":"10.11575/CDM.V15I1.62808","DOIUrl":"https://doi.org/10.11575/CDM.V15I1.62808","url":null,"abstract":"We show that there is a unique way to partition a $5times 5$ array of lattice points into restrictions of five circles. This result is extended to the $6times 5$ array, and used to show the optimality of a six-circle solution for the $6times 6$ array.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"15 1","pages":"12-21"},"PeriodicalIF":0.5,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41945735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-26DOI: 10.11575/CDM.V14I1.62676
Sizhong Zhou
A $P_{geq n}$-factor means a path factor with each component having at least $n$ vertices,where $ngeq2$ is an integer. A graph $G$ is called a $P_{geq n}$-factor deleted graph if $G-e$admits a $P_{geq n}$-factor for any $ein E(G)$. A graph $G$ is called a $P_{geq n}$-factorcovered graph if $G$ admits a $P_{geq n}$-factor containing $e$ for each $ein E(G)$. In thispaper, we first introduce a new parameter, i.e., sun toughness, which is denoted by $s(G)$. $s(G)$is defined as follows:$$s(G)=min{frac{|X|}{sun(G-X)}: Xsubseteq V(G), sun(G-X)geq2}$$if $G$ is not a complete graph, and $s(G)=+infty$ if $G$ is a complete graph, where $sun(G-X)$denotes the number of sun components of $G-X$. Then we obtain two sun toughness conditions for agraph to be a $P_{geq n}$-factor deleted graph or a $P_{geq n}$-factor covered graph. Furthermore,it is shown that our results are sharp.
{"title":"Sun toughness and $P_{geq3}$-factors in graphs","authors":"Sizhong Zhou","doi":"10.11575/CDM.V14I1.62676","DOIUrl":"https://doi.org/10.11575/CDM.V14I1.62676","url":null,"abstract":"A $P_{geq n}$-factor means a path factor with each component having at least $n$ vertices,where $ngeq2$ is an integer. A graph $G$ is called a $P_{geq n}$-factor deleted graph if $G-e$admits a $P_{geq n}$-factor for any $ein E(G)$. A graph $G$ is called a $P_{geq n}$-factorcovered graph if $G$ admits a $P_{geq n}$-factor containing $e$ for each $ein E(G)$. In thispaper, we first introduce a new parameter, i.e., sun toughness, which is denoted by $s(G)$. $s(G)$is defined as follows:$$s(G)=min{frac{|X|}{sun(G-X)}: Xsubseteq V(G), sun(G-X)geq2}$$if $G$ is not a complete graph, and $s(G)=+infty$ if $G$ is a complete graph, where $sun(G-X)$denotes the number of sun components of $G-X$. Then we obtain two sun toughness conditions for agraph to be a $P_{geq n}$-factor deleted graph or a $P_{geq n}$-factor covered graph. Furthermore,it is shown that our results are sharp.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46620847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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