A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour
{"title":"The site-perimeter of words","authors":"A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour","doi":"10.22108/TOC.2017.21465","DOIUrl":"https://doi.org/10.22108/TOC.2017.21465","url":null,"abstract":"","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"37-48"},"PeriodicalIF":0.4,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45417774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, E. M'{a}v{c}ajov'{a} and M. v{S}koviera proved that every bidirected Eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. This result shows the validity of Bouchet's nowhere zero conjecture for Eulerian bidirected graphs. In this paper we prove the same theorem in a different terminology and with a short and simple proof. More precisely, we prove that every Eulerian undirected graph which admits a zero-sum flow, admits a zero-sum $4$-flow. As a conclusion we obtain a shorter proof for the previously mentioned result of M'{a}v{c}ajov'{a} and v{S}koviera.
{"title":"A new proof of validity of Bouchet's conjecture on Eulerian bidirected graphs","authors":"N. Ghareghani","doi":"10.22108/TOC.2017.21362","DOIUrl":"https://doi.org/10.22108/TOC.2017.21362","url":null,"abstract":"Recently, E. M'{a}v{c}ajov'{a} and M. v{S}koviera proved that every bidirected Eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. This result shows the validity of Bouchet's nowhere zero conjecture for Eulerian bidirected graphs. In this paper we prove the same theorem in a different terminology and with a short and simple proof. More precisely, we prove that every Eulerian undirected graph which admits a zero-sum flow, admits a zero-sum $4$-flow. As a conclusion we obtain a shorter proof for the previously mentioned result of M'{a}v{c}ajov'{a} and v{S}koviera.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"31-35"},"PeriodicalIF":0.4,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45759654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a graph and $chi^{prime}_{aa}(G)$ denotes the minimum number of colors required for an acyclic edge coloring of $G$ in which no two adjacent vertices are incident to edges colored with the same set of colors. We prove a general bound for $chi^{prime}_{aa}(Gsquare H)$ for any two graphs $G$ and $H$. We also determine exact value of this parameter for the Cartesian product of two paths, Cartesian product of a path and a cycle, Cartesian product of two trees, hypercubes. We show that $chi^{prime}_{aa}(C_msquare C_n)$ is at most $6$ fo every $mgeq 3$ and $ngeq 3$. Moreover in some cases we find the exact value of $chi^{prime}_{aa}(C_msquare C_n)$.
{"title":"Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs","authors":"F. S. Mousavi, M. Noori","doi":"10.22108/TOC.2017.20988","DOIUrl":"https://doi.org/10.22108/TOC.2017.20988","url":null,"abstract":"Let $G$ be a graph and $chi^{prime}_{aa}(G)$ denotes the minimum number of colors required for an acyclic edge coloring of $G$ in which no two adjacent vertices are incident to edges colored with the same set of colors. We prove a general bound for $chi^{prime}_{aa}(Gsquare H)$ for any two graphs $G$ and $H$. We also determine exact value of this parameter for the Cartesian product of two paths, Cartesian product of a path and a cycle, Cartesian product of two trees, hypercubes. We show that $chi^{prime}_{aa}(C_msquare C_n)$ is at most $6$ fo every $mgeq 3$ and $ngeq 3$. Moreover in some cases we find the exact value of $chi^{prime}_{aa}(C_msquare C_n)$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"19-30"},"PeriodicalIF":0.4,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45759862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$. In this paper we give the sufficient condition for a graph $AG(R)$ to be complete. We characterize rings for which $AG(R)$ is a regular graph, we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex. Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph.
{"title":"On annihilator graph of a finite commutative ring","authors":"Sanghita Dutta, Chanlemki Lanong","doi":"10.22108/TOC.2017.20360","DOIUrl":"https://doi.org/10.22108/TOC.2017.20360","url":null,"abstract":"The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$. In this paper we give the sufficient condition for a graph $AG(R)$ to be complete. We characterize rings for which $AG(R)$ is a regular graph, we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex. Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"1-11"},"PeriodicalIF":0.4,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41657412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A graph $G$ is called a fractional $(k,n',m)$-critical deleted graph if any $n'$ vertices are removed from $G$ the resulting graph is a fractional $(k,m)$-deleted graph. In this paper, we prove that for integers $kge 2$, $n',mge0$, $nge8k+n'+4m-7$, and $delta(G)ge k+n'+m$, if $$|N_{G}(x)cup N_{G}(y)|gefrac{n+n'}{2}$$ for each pair of non-adjacent vertices $x$, $y$ of $G$, then $G$ is a fractional $(k,n',m)$-critical deleted graph. The bounds for neighborhood union condition, the order $n$ and the minimum degree $delta(G)$ of $G$ are all sharp.
{"title":"A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs","authors":"Yun Gao, M. Farahani, Wei Gao","doi":"10.22108/TOC.2017.20355","DOIUrl":"https://doi.org/10.22108/TOC.2017.20355","url":null,"abstract":"A graph $G$ is called a fractional $(k,n',m)$-critical deleted graph if any $n'$ vertices are removed from $G$ the resulting graph is a fractional $(k,m)$-deleted graph. In this paper, we prove that for integers $kge 2$, $n',mge0$, $nge8k+n'+4m-7$, and $delta(G)ge k+n'+m$, if $$|N_{G}(x)cup N_{G}(y)|gefrac{n+n'}{2}$$ for each pair of non-adjacent vertices $x$, $y$ of $G$, then $G$ is a fractional $(k,n',m)$-critical deleted graph. The bounds for neighborhood union condition, the order $n$ and the minimum degree $delta(G)$ of $G$ are all sharp.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"13-19"},"PeriodicalIF":0.4,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45056139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The set of all non-increasing non-negative integer sequences $pi=(d_1, d_2,ldots,d_n)$ is denoted by $NS_n$. A sequence $piin NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $pi$. The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$. The complete product split graph on $L + M$ vertices is denoted by $overline{S}_{L, M}=K_{L} vee overline{K}_{M}$, where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. Another split graph is denoted by $S_{L, M} = overline{S}_{r_{1}, s_{1}} veeoverline{S}_{r_{2}, s_{2}} vee cdots vee overline{S}_{r_{p}, s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$. A sequence $pi=(d_{1}, d_{2},ldots,d_{n})$ is said to be potentially $S_{L, M}$-graphic (respectively $overline{S}_{L, M}$)-graphic if there is a realization $G$ of $pi$ containing $S_{L, M}$ (respectively $overline{S}_{L, M}$) as a subgraph. If $pi$ has a realization $G$ containing $S_{L, M}$ on those vertices having degrees $d_{1}, d_{2},ldots,d_{L+M}$, then $pi$ is potentially $A_{L, M}$-graphic. A non-increasing sequence of non-negative integers $pi = (d_{1}, d_{2},ldots,d_{n})$ is potentially $A_{L, M}$-graphic if and only if it is potentially $S_{L, M}$-graphic. In this paper, we obtain the sufficient condition for a graphic sequence to be potentially $A_{L, M}$-graphic and this result is a generalization of that given by J. H. Yin on split graphs.
{"title":"The condition for a sequence to be potentially $A_{L, M}$- graphic","authors":"S. Pirzada, Bilal A. Chat","doi":"10.22108/TOC.2017.20361","DOIUrl":"https://doi.org/10.22108/TOC.2017.20361","url":null,"abstract":"The set of all non-increasing non-negative integer sequences $pi=(d_1, d_2,ldots,d_n)$ is denoted by $NS_n$. A sequence $piin NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $pi$. The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$. The complete product split graph on $L + M$ vertices is denoted by $overline{S}_{L, M}=K_{L} vee overline{K}_{M}$, where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. Another split graph is denoted by $S_{L, M} = overline{S}_{r_{1}, s_{1}} veeoverline{S}_{r_{2}, s_{2}} vee cdots vee overline{S}_{r_{p}, s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$. A sequence $pi=(d_{1}, d_{2},ldots,d_{n})$ is said to be potentially $S_{L, M}$-graphic (respectively $overline{S}_{L, M}$)-graphic if there is a realization $G$ of $pi$ containing $S_{L, M}$ (respectively $overline{S}_{L, M}$) as a subgraph. If $pi$ has a realization $G$ containing $S_{L, M}$ on those vertices having degrees $d_{1}, d_{2},ldots,d_{L+M}$, then $pi$ is potentially $A_{L, M}$-graphic. A non-increasing sequence of non-negative integers $pi = (d_{1}, d_{2},ldots,d_{n})$ is potentially $A_{L, M}$-graphic if and only if it is potentially $S_{L, M}$-graphic. In this paper, we obtain the sufficient condition for a graphic sequence to be potentially $A_{L, M}$-graphic and this result is a generalization of that given by J. H. Yin on split graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"21-27"},"PeriodicalIF":0.4,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42635835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a simple graph, and $G^{sigma}$ be an oriented graph of $G$ with the orientation $sigma$ and skew-adjacency matrix $S(G^{sigma})$. The $k-$th skew spectral moment of $G^{sigma}$, denoted by $T_k(G^{sigma})$, is defined as $sum_{i=1}^{n}( lambda_{i})^{k}$, where $lambda_{1}, lambda_{2},cdots, lambda_{n}$ are the eigenvalues of $G^{sigma}$. Suppose $G^{sigma_1}_{1}$ and $G^{sigma_2}_{2}$ are two digraphs. If there exists an integer $k$, $1 leq k leq n-1$, such that for each $i$, $0 leq i leq k-1$, $T_i(G^{sigma_1}_{1}) = T_i(G^{sigma_2}_{2})$ and $T_k(G^{sigma_1}_{1})
设$G$是一个简单图，$G^{sigma}$ $是$G$的一个有向图，有向$sigma$和斜邻接矩阵$S(G^{sigma})$ $。的k - th美元倾斜光谱时刻$ G ^{σ}$用T_k美元(G ^{σ})$被定义为美元sum_ {i = 1} ^ {n}(lambda_{我})^ {k} $,美元lambda_{1},lambda_ {2}, cdots,lambda_ {n} $的特征值是$ G ^{σ}$。假设$ G ^ {sigma_1} _{1} $和$ G ^ {sigma_2} _{2} $是两个有向图。如果存在整数k美元,1 leq k leq n - 1美元,每个这样我美元,0 leq我leq k - 1美元,T_i美元(G ^ {sigma_1} _{1}) =T_i (G ^ {sigma_2} _{2})和美元T_k美元(G ^ {sigma_1} _ {1}) < T_k (G ^ {sigma_ 2} _{2})美元然后写$ G ^ {sigma_1} _ {1} prec_ {T} G ^ {sigma_2} _{2} $。在本文中，我们确定了一些有向图的偏谱矩。我们还对一些关于偏谱矩的有向单环图进行了排序。
{"title":"On the skew spectral moments of graphs","authors":"F. Taghvaee, G. Fath-Tabar","doi":"10.22108/TOC.2017.20737","DOIUrl":"https://doi.org/10.22108/TOC.2017.20737","url":null,"abstract":"Let $G$ be a simple graph, and $G^{sigma}$ be an oriented graph of $G$ with the orientation $sigma$ and skew-adjacency matrix $S(G^{sigma})$. The $k-$th skew spectral moment of $G^{sigma}$, denoted by $T_k(G^{sigma})$, is defined as $sum_{i=1}^{n}( lambda_{i})^{k}$, where $lambda_{1}, lambda_{2},cdots, lambda_{n}$ are the eigenvalues of $G^{sigma}$. Suppose $G^{sigma_1}_{1}$ and $G^{sigma_2}_{2}$ are two digraphs. If there exists an integer $k$, $1 leq k leq n-1$, such that for each $i$, $0 leq i leq k-1$, $T_i(G^{sigma_1}_{1}) = T_i(G^{sigma_2}_{2})$ and $T_k(G^{sigma_1}_{1}) <T_k(G^{sigma_ 2}_{2})$ then we write $G^{sigma_1}_{1} prec_{T} G^{sigma_2}_{2}$. In this paper, we determine some of the skew spectral moments of oriented graphs. Also we order some oriented unicyclic graphs with respect to skew spectral moment.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"47-54"},"PeriodicalIF":0.4,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45734128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R$ be a commutative ring with identity. We use $varphi (R)$ to denote the comaximal ideal graph. The vertices of $varphi (R)$ are proper ideals of R which are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with planarity of line graph associated to $varphi (R)$.
{"title":"Some properties of comaximal ideal graph of a commutative ring","authors":"Z. Jafari, M. Azadi","doi":"10.22108/TOC.2017.20429","DOIUrl":"https://doi.org/10.22108/TOC.2017.20429","url":null,"abstract":"Let $R$ be a commutative ring with identity. We use $varphi (R)$ to denote the comaximal ideal graph. The vertices of $varphi (R)$ are proper ideals of R which are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with planarity of line graph associated to $varphi (R)$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"29-37"},"PeriodicalIF":0.4,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46880714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs, {em Australas. J. Combin.}, {bf 4} (1991) 229--235.), and present a similar method for constructing $t$-subset-regular self-complementary $k$-uniform hypergraphs of order $v$. As an application we show the existence of a new family of 2-subset-regular self-complementary 4-uniform hypergraphs with $v=16m+3$.
{"title":"A family of $t$-regular self-complementary $k$-hypergraphs","authors":"M. Ariannejad, M. Emami, O. Naserian","doi":"10.22108/TOC.2017.20363","DOIUrl":"https://doi.org/10.22108/TOC.2017.20363","url":null,"abstract":"We use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs, {em Australas. J. Combin.}, {bf 4} (1991) 229--235.), and present a similar method for constructing $t$-subset-regular self-complementary $k$-uniform hypergraphs of order $v$. As an application we show the existence of a new family of 2-subset-regular self-complementary 4-uniform hypergraphs with $v=16m+3$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"6 1","pages":"39-46"},"PeriodicalIF":0.4,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46547985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, by studying three $q$-Catalan identities given by Andrews, we arrive at a certain number of congruences. These congruences are all modulo $Phi_n(q)$, the $n$-th cyclotomic polynomial or the related functions and modulo $q$-integers.
{"title":"CONGRUENCES FROM Q-CATALAN IDENTITIES","authors":"Qing Zou","doi":"10.22108/TOC.2016.20358","DOIUrl":"https://doi.org/10.22108/TOC.2016.20358","url":null,"abstract":"In this paper, by studying three $q$-Catalan identities given by Andrews, we arrive at a certain number of congruences. These congruences are all modulo $Phi_n(q)$, the $n$-th cyclotomic polynomial or the related functions and modulo $q$-integers.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"5 1","pages":"57-67"},"PeriodicalIF":0.4,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68208555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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