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- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/06%3A_Graph_Theory/6.06%3A_Hamiltonian_Circuits_and_the_Traveling_Salesman_ProblemWith Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one wit...With Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one with lowest total weight. Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision without considering the consequences in the future.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/01%3A_Fundamentals/1.07%3A_The_Pigeonhole_PrincipleA key step in many proofs consists of showing that two possibly different values are in fact the same. The Pigeonhole principle can sometimes help with this.
- https://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/04%3A_Graph_Theory/4.02%3A_Hamiltonian_Circuits_and_the_Traveling_Salesman_ProblemWith Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one wit...With Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one with lowest total weight. Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision without considering the consequences in the future.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/01%3A_What_is_Combinatorics/1.03%3A_Some_Applications_of_Basic_Counting_PrinciplesIn this section, we explore the applications of the basic counting principles discussed in the previous section, one of which is the Pigeonhole Principle. The pigeonhole principle gets its name from t...In this section, we explore the applications of the basic counting principles discussed in the previous section, one of which is the Pigeonhole Principle. The pigeonhole principle gets its name from the idea of a grid of little boxes that might be used, for example, to sort mail, or as mailboxes for a group of people in an office. The boxes in such grids are sometimes called pigeonholes in analogy with stacks of boxes used to house homing pigeons when homing pigeons were used to carry messages
- https://math.libretexts.org/Courses/Chabot_College/Math_in_Society_(Zhang)/13%3A_Graph_Theory/13.06%3A_Hamiltonian_Circuits_and_the_Traveling_Salesman_ProblemWith Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one wit...With Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one with lowest total weight. Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision without considering the consequences in the future.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/04%3A_Systems_of_Distinct_Representatives/4.05%3A_Introduction_to_Graph_TheoryA cycle is a graph Cn on vertices v1,v2,…,vn with edges {vi,v1+(imodn)} for 1≤i≤n, and no other edges; this is a path in which the first and last vertices have ...A cycle is a graph Cn on vertices v1,v2,…,vn with edges {vi,v1+(imodn)} for 1≤i≤n, and no other edges; this is a path in which the first and last vertices have been joined by an edge. (Generally, we require that a cycle have at least three vertices. If a cycle has one vertex, there is an edge, called a loop, in which a single vertex serves as both endpoints.) The length of a path or cycle is the number of edges in the graph.
- https://math.libretexts.org/Courses/Northwest_Florida_State_College/MGF_1131%3A_Mathematics_in_Context/05%3A_Voting_and_Graph_Theory/5.09%3A_Hamiltonian_Circuits_and_the_Traveling_Salesman_ProblemWith Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one wit...With Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one with lowest total weight.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/03%3A_Graph_Theory/11%3A_Basics_of_Graph_Theory/11.03%3A_Deletion_Complete_Graphs_and_the_Handshaking_LemmaWe’ll begin this section by introducing a basic operation that can change a graph (or a multigraph, with or without loops) into a smaller graph: deletion. Then, we will define a very important family ...We’ll begin this section by introducing a basic operation that can change a graph (or a multigraph, with or without loops) into a smaller graph: deletion. Then, we will define a very important family of graphs, called complete graphs. Finally, we will introduce Euler's Handshaking Lemma.