scholarly journals Every toroidal graph without adjacent triangles is (4,1)*-choosable

2007 ◽  
Vol 155 (1) ◽  
pp. 74-78 ◽  
Author(s):  
Baogang Xu ◽  
Haihui Zhang
Keyword(s):  
Toroidal Graph

A new sufficient condition for a toroidal graph to be 4-choosable

2018 ◽  
Vol 341 (10) ◽  
pp. 2878-2882
Author(s):  
Jing Lv ◽  
Danjun Huang
Keyword(s):  
Sufficient Condition ◽  
Toroidal Graph

A BOUND ON THE BONDAGE NUMBER OF TOROIDAL GRAPHS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250046 ◽  
Author(s):  
JIANFENG HOU ◽  
GUIZHEN LIU
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Bondage Number ◽  
Toroidal Graphs ◽  
Toroidal Graph

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than γ(G), where γ(G) denote the domination number of G. Let G be a toroidal graph with maximum degree Δ(G). In this paper, we show that b(G) ≤ 9. Moveover, if Δ(G) ≠ 6, then b(G) ≤ 8.

scholarly journals INCOMPRESSIBLE SURFACES AND (1, 1)-KNOTS

2006 ◽  
Vol 15 (07) ◽  
pp. 935-948 ◽  
Author(s):  
MARIO EUDAVE-MUÑOZ
Keyword(s):  
Lens Space ◽  
Regular Neighborhood ◽  
Incompressible Surface ◽  
Special Position ◽  
Incompressible Surfaces ◽  
Toroidal Graph ◽  
Tunnel Number ◽  
Tunnel Number One Knots

Let M be S3, S1 × S2, or a lens space L(p, q), and let k be a (1, 1)-knot in M. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1, 1)-knots.

scholarly journals Bounding Tree-Width via Contraction on the Projective Plane and Torus.

10.37236/2534 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Evan Morgan ◽  
Bogdan Oporowski
Keyword(s):  
Projective Plane ◽  
Planar Graph ◽  
Tree Width ◽  
Toroidal Graph ◽  
Edge Partition

If $X$ is a collection of edges in a graph $G$, let $G/X$ denote the contraction of $X$. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective planar graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three. We prove that every toroidal graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three and four, respectively.

The Toughness of a Toroidal Graph with Connectivity 3 and Girth 6 is Exactly 1

2004 ◽  
Vol 20 (2) ◽  
pp. 181-183 ◽  
Author(s):  
Weiting Cao ◽  
Michael J. Pelsmajer
Keyword(s):  
Toroidal Graph

Every toroidal graph is acyclically 8-choosable

2013 ◽  
Vol 30 (2) ◽  
pp. 343-352 ◽  
Author(s):  
Jian Feng Hou ◽  
Gui Zhen Liu
Keyword(s):  
Toroidal Graph

ON (3, 1)*-CHOOSABILITY OF TOROIDAL GRAPHS

2009 ◽  
Vol 01 (02) ◽  
pp. 291-297 ◽  
Author(s):  
WEI DONG ◽  
BAOGANG XU
Keyword(s):  
Toroidal Graphs ◽  
Toroidal Graph

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v ∈ V(G). Given a list-assignment L and integer d ≥ 0, an (L, d)*-coloring of a graph G is a mapping ϕ that assigns a color ϕ(v) ∈ L(v) to each vertex v ∈ V(G) such that at most d neighbors of v receive color ϕ(v). A graph is called (k,d)*-choosable, if G admits an (L, d)*-coloring for every list assignment L with |L(v)| ≥ k for all v ∈ V(G). In this note, it is proved that every toroidal graph, which contains neither i-cycles nor j-cycles for any subset {i, j} ⊆ {3, 4, 6}, is (3, 1)*-choosable.

scholarly journals A Maximal Toroidal Graph which is not a Triangulation.

1973 ◽  
Vol 33 ◽  
pp. 108 ◽  
Author(s):  
F. Harary ◽  
P. C. Kainen ◽  
A. J. Schwenk
Keyword(s):  
Toroidal Graph
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