Constructions of cubic bipartite 3-connected graphs without Hamiltonian cycles

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.

Download Full-text

A note on maximal hamiltonian cycles in the square of a graph

Demonstratio Mathematica ◽

10.1515/dema-1988-0419 ◽

1988 ◽

Vol 21 (4) ◽

pp. 1089-1093

Author(s):

Tomasz Traczyk

Keyword(s):

Hamiltonian Cycles ◽

Square Of A Graph

Download Full-text

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

Journal of Physics Conference Series ◽

10.1088/1742-6596/1751/1/012023 ◽

2021 ◽

Vol 1751 ◽

pp. 012023

Author(s):

F C Puri ◽

Wamiliana ◽

M Usman ◽

Amanto ◽

M Ansori ◽

...

Keyword(s):

Connected Graphs

Download Full-text

Hamiltonicity of a class of toroidal graphs

Mathematica Slovaca ◽

10.1515/ms-2017-0367 ◽

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.

Download Full-text

On minimum algebraic connectivity of graphs whose complements are bicyclic

Open Mathematics ◽

10.1515/math-2019-0119 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1490-1502 ◽

Cited By ~ 1

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.

Download Full-text