Geodesic
On the girth of three-dimensional algebraically defined graphs with multiplicatively separable functions
Alex M. Kodess1 ; Brian G. Kronenthal2 ; Tony W.H. Wong2
1Farmingdale State College
2Kutztown University of Pennsylvania
The electronic journal of combinatorics, Tome 29 (2022) no. 4
Cet article a éte moissonné depuis la source The Electronic Journal of Combinatorics website

Voir la notice de l'article

For a field $\mathbb{F}$ and functions $f,g,h,j\colon\mathbb{F}\to \mathbb{F}$, we define $\Gamma_\mathbb{F}(f(X)h(Y),g(X) j(Y))$ to be a bipartite graph where each partite set is a copy of $\mathbb{F}^3$, and a vertex $(a,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x,x_2,x_3]$ in the second partite set if and only if \[a_2+x_2=f(a)h(x) \quad \text{and} \quad a_3+x_3=g(a)j(x).\] In this paper, we completely classify all such graphs by girth in the case $h=j$ (subject to some mild restrictions on $h$). We also present a partial classification when $h\neq j$ and provide some applications.
DOI : 10.37236/9749
Classification : 05C25, 05C38
Mots-clés : algebraically defined graphs, girth

Alex M. Kodess  1   ; Brian G. Kronenthal  2   ; Tony W.H. Wong  2

1 Farmingdale State College
2 Kutztown University of Pennsylvania 
@article{10_37236_9749,
     author = {Alex M. Kodess and Brian G. Kronenthal and Tony W.H. Wong},
     title = {On the girth of three-dimensional algebraically defined graphs with multiplicatively separable functions},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {4},
     doi = {10.37236/9749},
     zbl = {1506.05091},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9749/}
}
TY  - JOUR
AU  - Alex M. Kodess
AU  - Brian G. Kronenthal
AU  - Tony W.H. Wong
TI  - On the girth of three-dimensional algebraically defined graphs with multiplicatively separable functions
JO  - The electronic journal of combinatorics
PY  - 2022
VL  - 29
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.37236/9749/
DO  - 10.37236/9749
ID  - 10_37236_9749
ER  - 
%0 Journal Article
%A Alex M. Kodess
%A Brian G. Kronenthal
%A Tony W.H. Wong
%T On the girth of three-dimensional algebraically defined graphs with multiplicatively separable functions
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/9749/
%R 10.37236/9749
%F 10_37236_9749
Alex M. Kodess; Brian G. Kronenthal; Tony W.H. Wong. On the girth of three-dimensional algebraically defined graphs with multiplicatively separable functions. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/9749

Cité par Sources :