Search results
Results from the WOW.Com Content Network
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.
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 ...
In more colloquial terms, in a party of people some of whom shake hands, an even number of people must have shaken an odd number of other people's hands; for this reason, the result is known as the handshaking lemma. To prove this by double counting, let () be the degree of vertex . The number of vertex-edge incidences in the graph may be ...
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 ...
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 ...
The edges of this matching represent paths in the original graph, whose union forms the desired T-join. Both constructing the complete graph, and then finding a matching in it, can be done in O(n 3) computational steps. For the route inspection problem, T should be chosen as the set of all odd
A graph meeting the conditions of Ore's theorem, and a Hamiltonian cycle in it. There are two vertices with degree less than n/2 in the center of the drawing, so the conditions for Dirac's theorem are not met. However, these two vertices are adjacent, and all other pairs of vertices have total degree at least seven, the number of vertices.
Therefore, the vertex of G corresponding to the outer area has an odd degree. But it is known (the handshaking lemma) that in a finite graph there is an even number of vertices with odd degree. Therefore, the remaining graph, excluding the outer area, has an odd number of vertices with odd degree corresponding to members of T.