scholarly journals Counting Hamiltonian Cycles in 2-Tiled Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 693
Author(s):  
Alen Vegi Kalamar ◽  
Tadej Žerak ◽  
Drago Bokal
Keyword(s):  
Hamiltonian Cycles ◽  
Critical Graphs ◽  
Connected Graphs ◽  
Forbidden Minors

In 1930, Kuratowski showed that K3,3 and K5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol showed that there are infinitely many k-crossing-critical graphs for any k≥2, even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Pantić, Kwong, Doroslovački, and Pantić.

scholarly journals Hamiltonian cycles in n-factor-critical graphs

2001 ◽  
Vol 240 (1-3) ◽  
pp. 71-82 ◽  
Author(s):  
Ken-ichi Kawarabayashi ◽  
Katsuhiro Ota ◽  
Akira Saito
Keyword(s):  
Hamiltonian Cycles ◽  
Critical Graphs

Hamiltonian Cycles in Critical Graphs with Large Maximum Degree

2016 ◽  
Vol 32 (5) ◽  
pp. 2019-2028 ◽  
Author(s):  
Rong Luo ◽  
Zhengke Miao ◽  
Yue Zhao
Keyword(s):  
Maximum Degree ◽  
Hamiltonian Cycles ◽  
Critical Graphs ◽  
Large Maximum

scholarly journals Improvement on the Crossing Number of Crossing-Critical Graphs

2020 ◽  
Author(s):  
János Barát ◽  
Géza Tóth
Keyword(s):  
Crossing Number ◽  
Critical Graphs ◽  
Minimum Number ◽  
Edge Crossings

AbstractThe crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

scholarly journals The Formula to Count The Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops

2021 ◽  
Vol 1751 ◽  
pp. 012023
Author(s):  
F C Puri ◽  
Wamiliana ◽  
M Usman ◽  
Amanto ◽  
M Ansori ◽  
...  
Keyword(s):  
Connected Graphs

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

scholarly journals Dynamics of epidemic spreading on connected graphs

2021 ◽  
Vol 82 (6) ◽  
Author(s):  
Christophe Besse ◽  
Grégory Faye
Keyword(s):  
Epidemic Spreading ◽  
Connected Graphs
Sign in / Sign up

Export Citation Format

Share Document