University of Isfahan Transactions on Combinatorics 2251-8657 6 2 2017 06 01 On numerical semigroups with embedding dimension three 1 6 20736 10.22108/toc.2017.20736 EN Ali Mahdavi Amirkabir University of Technology Farhad Rahmati Amirkabir University of Technology Journal Article 2016 07 11 Let $f\neq1,3$ 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.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 6 2 2017 06 01 Full edge-friendly index sets of complete bipartite graphs 7 17 20739 10.22108/toc.2017.20739 EN Wai Chee Shiu Hong Kong Baptist University Journal Article 2016 01 20 ‎‎Let $G=(V,E)$ 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.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 6 2 2017 06 01 Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs 19 30 20988 10.22108/toc.2017.20988 EN Fatemeh Sadat Mousavi University of Zanjan Massomeh Noori University of Zanjan Journal Article 2016 01 13 ‎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 $\chi^{\prime}_{aa}(G\square H)$ 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.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 6 2 2017 06 01 A new proof of validity of Bouchet's conjecture on Eulerian bidirected graphs 31 35 21362 10.22108/toc.2017.21362 EN Narges Ghareghani University of Tehran Journal Article 2016 12 07 Recently, 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.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 6 2 2017 06 01 The site-perimeter of words 37 48 21465 10.22108/toc.2017.21465 EN Aubrey Blecher University of the Witwatersrand 0000-0003-2487-3220 Charlotte Brennan 1 Jan Smuts Avenue Arnold Knopfmacher University of the Witwatersrand 0000-0003-1962-043X Toufik Mansour University of the Witwatersrand Journal Article 2016 01 26 We 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