We show that the mathematical proof of the four color theorem yields a perfect interpretation of the Standard Model of particle physics. The steps of the proof enable us to construct the t-Riemann surface and particle frame which forms the gauge. We specify well-defined rules to match the Standard Model in a one-to-one correspondence with the topological and algebraic structure of the particle frame. This correspondence is exact - it only allows the particles and force fields to have the observable properties of the Standard Model, giving us a Grand Unified Theory. In this paper, we concentrate on explicitly specifying the quarks, gauge vector bosons, the Standard Model scalar Higgs boson and the weak force field. Using all the specifications of our mathematical model, we show how to calculate the values of the Weinberg and Cabibbo angles on the particle frame. Finally, we present our prediction of the Higgs boson mass M = 126 GeV, as a direct consequence of the proof of the four color theorem.

We show that the mathematical proof of the four colour theorem directly implies the existence of the standard model together with quantum gravity in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein's law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with 't Hooft's table. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles. We show how to calculate Einstein's cosmological constant using the grand unified theory. Using the topological properties of the gauge, we calculate the exact percentages of ordinary baryonic matter, dark matter and dark energy in the universe. These values are in perfect agreement with the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations. Thus dark matter, dark energy and the cosmological constant are intrinsic properties of the gauge in the grand unified theory.

We present a new polynomial-time algorithm for finding maximal independent sets in graphs. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and minimum vertex degrees of the corresponding Ramsey graphs. The algorithm finds a maximum independent set in all known examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cannot find a maximum independent set. The algorithm is demonstrated by finding maximum independent sets for several famous graphs, including two large benchmark graphs with hidden maximum independent sets. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.

We present a new polynomial-time algorithm for finding proper m-colorings of the vertices of a graph. We prove that every graph with n vertices and maximum vertex degree Delta must have chromatic number Chi(G) less than or equal to Delta+1 and that the algorithm will always find a proper m-coloring of the vertices of G with m less than or equal to Delta+1. Furthermore, we prove that this condition is the best possible in terms of n and Delta by explicitly constructing graphs for which the chromatic number is exactly Delta+1. In the special case when G is a connected simple graph and is neither an odd cycle nor a complete graph, we show that the algorithm will always find a proper m-coloring of the vertices of G with m less than or equal to Delta. In the process, we obtain a new constructive proof of Brooks' famous theorem of 1941. For all known examples of graphs, the algorithm finds a proper m-coloring of the vertices of the graph G for m equal to the chromatic number Chi(G). In view of the importance of the P versus NP question, we ask: does there exist a graph G for which this algorithm cannot find a proper m-coloring of the vertices of G with m equal to the chromatic number Chi(G)? The algorithm is demonstrated with several examples of famous graphs, including a proper four-coloring of the map of India and two large Mycielski benchmark graphs with hidden minimum vertex colorings. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.

We present a new polynomial-time algorithm for finding minimal vertex covers in graphs. The algorithm finds a minimum vertex cover in all known examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cannot find a minimum vertex cover. The algorithm is demonstrated by finding minimum vertex covers for several famous graphs, including two large benchmark graphs with hidden minimum vertex covers. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.

We present a new polynomial-time algorithm for finding Hamiltonian circuits in graphs. It is shown that the algorithm always finds a Hamiltonian circuit in graphs that have at least three vertices and minimum degree at least half the total number of vertices. In the process, we also obtain a constructive proof of Dirac's famous theorem of 1952, for the first time. The algorithm finds a Hamiltonian circuit (respectively, tour) in all known examples of graphs that have a Hamiltonian circuit (respectively, tour). In view of the importance of the P versus NP question, we ask: does there exist a graph that has a Hamiltonian circuit (respectively, tour) but for which this algorithm cannot find a Hamiltonian circuit (respectively, tour)? The algorithm is implemented in C++ and the program is demonstrated with several examples.

Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. It is being actively used in fields as varied as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove fundamental results in other areas of pure mathematics. This book, besides giving a general outlook of these facts, includes new graph theoretical proofs of Fermat's Little Theorem and the Nielson-Schreier Theorem. New applications to DNA sequencing (the SNP assembly problem) and computer network security (worm propagation) using minimum vertex covers in graphs are discussed. We also show how to apply edge coloring and matching in graphs for scheduling (the timetabling problem) and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. Finally, we revisit the classical problem of finding re-entrant knight's tours on a chessboard using Hamiltonian circuits in graphs.

We show how the grand unified theory based on the proof of the four color theorem can be obtained entirely in terms of the Poincare group of isometries of space and time. Electric and gauge charges of all the particles of the standard model can now be interpreted as elements of the Poincare group. We define the space and time chiralities of all spin 1/2 fermions in agreement with Dirac's relativistic wave equation. All the particles of the standard model now correspond to irreducible representations of the Poincare group according to Wigner's classification. Finally, we construct the Steiner system of fermions and show how the Mathieu group acts as the group of symmetries of the fundamental building blocks of matter.

We present a new polynomial-time algorithm for finding maximal cliques in graphs. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and minimum vertex degrees of the corresponding Ramsey graphs. The algorithm finds a maximum clique in all known examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cannot find a maximum clique. The algorithm is demonstrated by finding maximum cliques for several famous graphs, including two large benchmark graphs with hidden maximum cliques. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.

We present a new polynomial-time algorithm for determining whether two given graphs are isomorphic or not. We prove that the algorithm is necessary and sufficient for solving the Graph Isomorphism Problem in polynomial-time, thus showing that the Graph Isomorphism Problem is in P. The semiotic theory for the recognition of graph structure is used to define a canonical form of the sign matrix of a graph. We prove that the canonical form of the sign matrix is uniquely identifiable in polynomial-time for isomorphic graphs. The algorithm is demonstrated by solving the Graph Isomorphism Problem for many of the hardest known examples. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.

We present a new proof of the famous four colour theorem using algebraic and topological methods. This proof was first announced by the Canadian Mathematical Society in 2000 and subsequently published by Orient Longman and Universities Press of India in 2008. Recent research in physics shows that this proof directly implies the Grand Unification of the Standard Model with Quantum Gravity in its physical interpretation and conversely the existence of the standard model of particle physics shows that nature applies this proof of the four colour theorem at the most fundamental level.

This text offers the most comprehensive and up-to-date presentation available on the fundamental topics in graph theory. It develops a thorough understanding of the structure of graphs, the techniques used to analyze problems in graph theory and the uses of graph theoretical algorithms in mathematics, engineering and computer science. There are many new topics in this book that have not appeared before in print: new proofs of various classical theorems, signed degree sequences, criteria for graphical sequences, eccentric sequences, matching and decomposition of planar graphs into trees. Scores in digraphs appear for the first time and include new results due to Pirzada. The climax of the book is a new proof of the famous four colour theorem due to Dharwadker.