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It has n 2 + 2n + 1 vertices: n 2 formed from the product of a leaf in both factors, 2n from the product of a leaf in one factor and the hub in the other factor, and one remaining vertex formed from the product of the two hubs. Each leaf-hub product vertex in G dominates exactly n of the leaf-leaf vertices, so n leaf-hub vertices are needed to ...
Switching {X,Y} in a graph. A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent ...
Clique-sums are important in graph structure theory, where they are used to characterize certain families of graphs as the graphs formed by clique-sums of simpler graphs. The first result of this type [2] was a theorem of Wagner (1937), who proved that the graphs that do not have a five-vertex complete graph as a minor are the 3-clique-sums of ...
The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f(v) to each vertex v, a graph G is f-choosable (or f-list-colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to ...
Less commonly (though more consistent with the general definition of union in mathematics) the union of two graphs is defined as the graph (V 1 ∪ V 2, E 1 ∪ E 2). graph intersection: G 1 ∩ G 2 = (V 1 ∩ V 2, E 1 ∩ E 2); [1] graph join: . Graph with all the edges that connect the vertices of the first graph with the vertices of the ...
The inequality χ(G × H) ≤ min {χ(G), χ(H)} is easy: if G is k-colored, one can k-color G × H by using the same coloring for each copy of G in the product; symmetrically if H is k-colored. Thus, Hedetniemi's conjecture amounts to the assertion that tensor products cannot be colored with an unexpectedly small number of colors.
In general, a subdivision of a graph G (sometimes known as an expansion [2]) is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints {u,v } yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, {u,w } and {w,v }. For directed edges, this operation shall ...
The Cartesian product of K 2 and a path graph is a ladder graph. The Cartesian product of two path graphs is a grid graph. The Cartesian product of n edges is a hypercube: =. Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q j = Q i+j. The Cartesian product of two median graphs is another median graph.