University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
6
2
2017
06
01
On numerical semigroups with embedding dimension three
1
6
EN
Farhad
Rahmati
Amirkabir University of Technology
frahmati@aut.ac.ir
Ali
Mahdavi
Amirkabir University of Technology
a_mahdavi@aut.ac.ir
Let $fneq1,3$ 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.
Frobenius number,Frobenius vector,Numerical semigroup,simplicial affine semigroup
http://toc.ui.ac.ir/article_20736.html
http://toc.ui.ac.ir/article_20736_f17d7b141a073f2b8afd89a9f3d6327d.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
6
2
2017
06
01
Full edge-friendly index sets of complete bipartite graphs
7
17
EN
Wai Chee
Shiu
Hong Kong Baptist University
wcshiu@hkbu.edu.hk
Let $G=(V,E)$ be a simple graph. An edge labeling $f:Eto {0,1}$ induces a vertex labeling $f^+:VtoZ_2$ defined by $f^+(v)equiv sumlimits_{uvin E} f(uv)pmod{2}$ for each $v in V$, where $Z_2={0,1}$ is the additive group of order 2. For $iin{0,1}$, 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.
Full edge-friendly index sets,edge-friendly index,edge-friendly labeling,complete bipartite graph
http://toc.ui.ac.ir/article_20739.html
http://toc.ui.ac.ir/article_20739_045d1f054d5254d803c8df777bac0d71.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
6
2
2017
06
01
Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs
19
30
EN
Fatemeh Sadat
Mousavi
University of Zanjan
fmousavi@znu.ac.ir
Massomeh
Noori
University of Zanjan
mnouri@znu.ac.ir
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 $chi^{prime}_{aa}(C_msquare C_n)$.
Acyclic edge coloring,adjacent vertex distinguishing acyclic edge coloring,adjacent vertex distinguishing acyclic edge chromatic number
http://toc.ui.ac.ir/article_20988.html
http://toc.ui.ac.ir/article_20988_dc6050dc4f36dd95fd12e657ff895814.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
6
2
2017
06
01
A new proof of validity of Bouchet's conjecture on Eulerian bidirected graphs
31
35
EN
Narges
Ghareghani
University of Tehran
ghareghani@ipm.ir
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.
Nowhere zero flow in bidirected graphs,zero-sum flow,Eulerian graphs
http://toc.ui.ac.ir/article_21362.html
http://toc.ui.ac.ir/article_21362_d500cc0139aaa06f9bbb49637cdb9ec2.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
6
2
2017
06
01
The site-perimeter of words
37
48
EN
Charlotte
Brennan
1 Jan Smuts Avenue
charlotte.brennan@wits.ac.za
Aubrey
Blecher
University of the Witwatersrand
aubrey.blecher@wits.ac.za
Arnold
Knopfmacher
University of the Witwatersrand
arnold.knopfmacher@wits.ac.za
Toufik
Mansour
University of the Witwatersrand
toufik@math.haifa.ac.il
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.
words,bargraphs,site-perimeter,generating functions
http://toc.ui.ac.ir/article_21465.html
http://toc.ui.ac.ir/article_21465_6b2d0d7534fbdaeab5e1760bad7055c7.pdf