2017
6
2
2
0
On numerical semigroups with embedding dimension three
2
2
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.
1

1
6


Farhad
Rahmati
Amirkabir University of Technology
Amirkabir University of Technology
Iran
frahmati@aut.ac.ir


Ali
Mahdavi
Amirkabir University of Technology
Amirkabir University of Technology
Iran
a_mahdavi@aut.ac.ir
Frobenius number
Frobenius vector
Numerical semigroup
simplicial affine semigroup
Full edgefriendly index sets of complete bipartite graphs
2
2
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 edgefriendly if $e_f(1)e_f(0)le 1$. $I_f(G)=v_f(1)v_f(0)$ is called the edgefriendly index of $G$ under an edgefriendly labeling $f$. The full edgefriendly index set of a graph $G$ is the set of all possible edgefriendly indices of $G$. Full edgefriendly index sets of complete bipartite graphs will be determined.
1

7
17


Wai Chee
Shiu
Hong Kong Baptist University
Hong Kong Baptist University
Hong Kong
wcshiu@hkbu.edu.hk
Full edgefriendly index sets
edgefriendly index
edgefriendly labeling
complete bipartite graph
Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs
2
2
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 $chi^{prime}_{aa}(C_msquare C_n)$.
1

19
30


Fatemeh Sadat
Mousavi
University of Zanjan
University of Zanjan
Iran
fmousavi@znu.ac.ir


Massomeh
Noori
University of Zanjan
University of Zanjan
Iran
mnouri@znu.ac.ir
Acyclic edge coloring
adjacent vertex distinguishing acyclic edge coloring
adjacent vertex distinguishing acyclic edge chromatic number
A new proof of validity of Bouchet's conjecture on Eulerian bidirected graphs
2
2
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 zerosum flow, admits a zerosum $4$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.
1

31
35


Narges
Ghareghani
University of Tehran
University of Tehran
Iran
ghareghani@ipm.ir
Nowhere zero flow in bidirected graphs
zerosum flow
Eulerian graphs
The siteperimeter of words
2
2
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 columnconvex 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 siteperimeter, which is the number of nearestneighbour cells outside the boundary of the polyomino. The generating function that counts the siteperimeter of words is obtained explicitly. From a functional equation we find the average siteperimeter 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 siteperimeters.
1

37
48


Charlotte
Brennan
1 Jan Smuts Avenue
1 Jan Smuts Avenue
South Africa
charlotte.brennan@wits.ac.za


Aubrey
Blecher
University of the Witwatersrand
University of the Witwatersrand
South Africa
aubrey.blecher@wits.ac.za


Arnold
Knopfmacher
University of the Witwatersrand
University of the Witwatersrand
South Africa
arnold.knopfmacher@wits.ac.za


Toufik
Mansour
University of the Witwatersrand
University of the Witwatersrand
United States of America
toufik@math.haifa.ac.il
words
bargraphs
siteperimeter
generating functions