The Fascinating World of Graph Theory

@inproceedings{Benjamin2015TheFW,
  title={The Fascinating World of Graph Theory},
  author={Arthur T. Benjamin and Gary Chartrand and Ping Zhang},
  year={2015}
}
The fascinating world of graph theory goes back several centuries and revolves around the study of graphsmathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematicsand some of its most famous problems. For example, what is the shortest route for a traveling salesman seeking to visit a number of cities in one trip? What is the least… 
HAMILTONICITY OF CAMOUFLAGE GRAPHS
  • S. K.
  • Computer Science
    International Journal of Engineering Applied Sciences and Technology
  • 2021
TLDR
The result proved that “Every connected vertex-transitive simple graph has a Hamilton path” shows a significant improvement over the previous efforts by L.Babai and L.Lovasz.
On Eulerian Irregularity and Decompositions in Graphs
For a nontrivial connected graph G of size m, it is shown that ( m+1 2 ) ≤ EI(G) ≤ 2 ( m+1 2 ) and that EI(G) = 2 ( m+1 2 ) if and only if G is a tree of size m. A necessary and sufficient condition
Collatz Sequences in the Light of Graph Theory
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that
Homometric Number of a Graph and Some Related Concepts
TLDR
It is proved that the homometric number of the Cartesian product of two graphs is at least twice the product of the homomet numbers of the individual graphs.
Business Network Analytics: From Graphs to Supernetworks
  • P. Moscato
  • Computer Science
    Business and Consumer Analytics: New Ideas
  • 2019
TLDR
This chapter discusses issues related to the computational complexity of some problems associated with data analytics and provides a survey on recent applications and new algorithmic approaches for data analytics.
A BIPARTITE GRAPH ASSOCIATED WITH IRREDUCIBLE ELEMENTS AND GROUP OF UNITS IN Z n
A nonzero nonunit a of a ring R is called an irreducible element if, for some b, c ∈ R, a = bc implies that either b or c (not both) is a unit. We construct a bipartite graph in which the union of
Puzzles and Spatial Reasoning
A main tenet of this book is that the idea-structure for many puzzles typically originates in the imagination (a right-hemispheric function) and then migrates, via a cognitive flow, to embed itself
The Mathematical Mind
As argued throughout this book, puzzles have played as much a role as any other human artifact, mental tool, or device in human history as sparks for discovery. The Ahmes Papyrus is more than a
Reviews
TLDR
This note assumes that the definition of ‘fascinating’ is some convex combination of these qualities, and gives a proof of the following result.
Puzzles and Mathematics
The English word puzzle covers a broad range of meanings, alluding to everything from riddles and crosswords to Sudoku, optical illusions, and difficult conundrums in advanced mathematics. As a
...
1
2
3
4
...

References

SHOWING 1-10 OF 79 REFERENCES
ON PROPERTIES OF A WELL‐KNOWN GRAPH OR WHAT IS YOUR RAMSEY NUMBER?
TLDR
This paper considers the collaboration graph G as a graph by replacing each (hyper)edge X by the complete graph K(X) on X, and considers the various extremal problems for G the authors will consider.
A Theory of Graphs
TLDR
The theory of graphs has broad and important applications, because so many things can be modeled by graphs, and various puzzles and games are solved easily if a little graph theory is applied.
Every Planar Map Is Four Colorable
As has become standard, the four color map problem will be considered in the dual sense as the problem of whether the vertices of every planar graph (without loops) can be colored with at most four
On the History of the Minimum Spanning Tree Problem
TLDR
There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
Four Colors Suffice: How the Map Problem Was Solved
On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound
A short proof of the factor theorem for finite graphs
We define a graph as a set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices,
Vertex colouring edge partitions
TLDR
It is shown that the edges of every graph can be partitioned into 4 sets such that the resultant multisets give a vertex colouring of G.
Cycle decompositions V: complete graphs into cycles of arbitrary lengths
We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1,..., mt if and only if n is odd, 3≤mi≤n for i=1,..., t, and. We also show that the complete graph
Linear cellular automata and the garden-of-eden
TLDR
A solution to the All-Ones Problem can be described by a subset of all squares (namely a set of squares whose buttons when pressed in an arbitrary order will render all lights on) rather than a sequence.
The Truth about Königsberg
Brian Hopkins (bhopkins@spc.edu) is an assistant professor at St. Peter’s College, a Jesuit liberal arts college in Jersey City, New Jersey. He received his Ph.D. from the University of Washington
...
1
2
3
4
5
...