University of IsfahanTransactions on Combinatorics2251-86576220170601On numerical semigroups with embedding dimension three162073610.22108/toc.2017.20736ENAliMahdaviAmirkabir University of TechnologyFarhadRahmatiAmirkabir University of TechnologyJournal Article20160711Let $f\neq1,3$ be a positive integer. We prove that there exists a numerical semigroup $S$ with embedding dimension three such that $f$ is the Frobenius number of $S$. We also show that the same fact holds for affine semigroups in higher dimensional monoids.https://toc.ui.ac.ir/article_20736_f17d7b141a073f2b8afd89a9f3d6327d.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576220170601Full edge-friendly index sets of complete bipartite graphs7172073910.22108/toc.2017.20739ENWai CheeShiuHong Kong Baptist UniversityJournal Article20160120Let $G=(V,E)$ be a simple graph. An edge labeling $f:E\to \{0,1\}$ induces a vertex labeling $f^+:V\to Z_2$ defined by $f^+(v)\equiv \sum\limits_{uv\in E} f(uv)\pmod{2}$ for each $v \in V$, where $Z_2=\{0,1\}$ is the additive group of order 2. For $i\in\{0,1\}$, let $e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$. A labeling $f$ is called edge-friendly if $|e_f(1)-e_f(0)|\le 1$. $I_f(G)=v_f(1)-v_f(0)$ is called the edge-friendly index of $G$ under an edge-friendly labeling $f$. The full edge-friendly index set of a graph $G$ is the set of all possible edge-friendly indices of $G$. Full edge-friendly index sets of complete bipartite graphs will be determined.https://toc.ui.ac.ir/article_20739_045d1f054d5254d803c8df777bac0d71.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576220170601Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs19302098810.22108/toc.2017.20988ENFatemeh SadatMousaviUniversity of ZanjanMassomehNooriUniversity of ZanjanJournal Article20160113Let $G$ 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 $\chi^{\prime}_{aa}(G\square H)$ 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 $\chi^{\prime}_{aa}(C_m\square C_n)$ is at most $6$ fo every $m\geq 3$ and $n\geq 3$. Moreover in some cases we find the exact value of $\chi^{\prime}_{aa}(C_m\square C_n)$.https://toc.ui.ac.ir/article_20988_dc6050dc4f36dd95fd12e657ff895814.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576220170601A new proof of validity of Bouchet's conjecture on Eulerian bidirected graphs31352136210.22108/toc.2017.21362ENNargesGhareghaniUniversity of TehranJournal Article20161207Recently, E. Máčajová and M. Škoviera 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 Máčajová and M. Škoviera.https://toc.ui.ac.ir/article_21362_d500cc0139aaa06f9bbb49637cdb9ec2.pdfUniversity of IsfahanTransactions on Combinatorics2251-86576220170601The site-perimeter of words37482146510.22108/toc.2017.21465ENAubreyBlecherUniversity of the Witwatersrand0000-0003-2487-3220CharlotteBrennan1 Jan Smuts AvenueArnoldKnopfmacherUniversity of the Witwatersrand0000-0003-1962-043XToufikMansourUniversity of the WitwatersrandJournal Article20160126We define $[k]=\{1, 2, 3,\ldots,k\}$ to be a (totally ordered) {\em alphabet} on $k$ letters. A {\em word} $w$ of length $n$ on the alphabet $[k]$ is an element of $[k]^n$. A word can be represented by a bargraph which is a family of column-convex polyominoes whose lower edge lies on the $x$-axis and in which the height of the $i$-th column in the bargraph equals the size of the $i$-th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the site-perimeter, which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the site-perimeter of words is obtained explicitly. From a functional equation we find the average site-perimeter of words of length $n$ over the alphabet $[k]$. We also show how these statistics may be obtained using a direct counting method and obtain the minimum and maximum values of the site-perimeters.https://toc.ui.ac.ir/article_21465_6b2d0d7534fbdaeab5e1760bad7055c7.pdf