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The simplest Loupekine snarks are obtained by constructing three blocks from three copies of the Petersen graph, connecting them by pairs of edges into a cycle of three blocks, and using a central graph consisting of a three-leaf star. There are two graphs of this type, depending on how the pairs of edges connecting consecutive blocks are chosen.
If G is a graph, the line graph L(G) has a vertex for each edge of G, and an edge for each pair of adjacent edges in G. Thus, the chromatic number of L(G) equals the chromatic index of G. If G is bipartite, the cliques in L(G) are exactly the sets of edges in G sharing a common endpoint. Now Kőnig's line coloring theorem, stating that the ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
The edge-connectivity version of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two distinct vertices. Then the size of the minimum edge cut for x and y (the minimum number of edges whose removal disconnects x and y) is equal to the maximum number of pairwise edge-disjoint paths from x to y.
A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and ...
The edge boundary is the set of edges with one endpoint in the inner boundary and one endpoint in the outer boundary. [ 1 ] These boundaries and their sizes are particularly relevant for isoperimetric problems in graphs , separator theorems , minimum cuts , expander graphs , and percolation theory .
A bipartite graph has its vertices split into two subsets, with each edge having one endpoint in each subset. It follows from the same double counting argument that, in each subset, the sum of degrees equals the number of edges in the graph. In particular, both subsets have equal degree sums. [21]
All remaining edges of the complete graph have distances given by the shortest paths in this subgraph. Then the minimum spanning tree will be given by the path, of length n − 1 , and the only two odd vertices will be the path endpoints, whose perfect matching consists of a single edge with weight approximately n /2 .