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The undirected route inspection problem can be solved in polynomial time by an algorithm based on the concept of a T-join.Let T be a set of vertices in a graph. An edge set J is called a T-join if the collection of vertices that have an odd number of incident edges in J is exactly the set T.
The odd graph = (,) In the mathematical field of graph theory, the odd graphs are a family of symmetric graphs defined from certain set systems. They include and generalize the Petersen graph. The odd graphs have high odd girth, meaning that they contain long odd-length cycles but no short ones.
Alternatively, it is possible to use mathematical induction to prove the degree sum formula, [2] or to prove directly that the number of odd-degree vertices is even, by removing one edge at a time from a given graph and using a case analysis on the degrees of its endpoints to determine the effect of this removal on the parity of the number of ...
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
The degree sequence problem is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some simple ...
As the Haus vom Nikolaus puzzle has two odd-degree vertices (orange), the trail must start at one and end at the other. A variant with four odd-degree vertices has no solution. If there are no odd-degree vertices, the trail can start anywhere and forms an Eulerian cycle. Loose ends are considered vertices of degree 1. The graph must also be ...
The Erdős–Gallai theorem is a result in graph theory, a branch of combinatorial mathematics. It provides one of two known approaches to solving the graph realization problem, i.e. it gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph. A sequence obeying these ...
But a polynomial of odd degree has an odd number of roots (fundamental theorem of algebra); Therefore some of them must be real. This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same multiplicity (and this lemma is not hard to prove).