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":null,"pages":null},"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":null,"pages":null},"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 : 2020-04-19DOI: 10.55016/ojs/cdm.v17i2.70232
A. D. Forbes, T. Griggs
The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper, we solve the design spectrum problem for the remaining five graphs with three possible exceptions.
{"title":"Designs for graphs with six vertices and ten edges - II","authors":"A. D. Forbes, T. Griggs","doi":"10.55016/ojs/cdm.v17i2.70232","DOIUrl":"https://doi.org/10.55016/ojs/cdm.v17i2.70232","url":null,"abstract":"The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper, we solve the design spectrum problem for the remaining five graphs with three possible exceptions.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141211310","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":null,"pages":null},"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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