Cartesian product of graphs

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In graph theory, the cartesian product G ◻ H of graphs G and H is a graph such that

• the vertex set of G ◻ H is the cartesian product V(G) × V(H); and
• any two vertices (u,u') and (v,v') are adjacent in G ◻ H if and only if either
  • u = v and u' is adjacent with v' , or
  • u' = v' and u is adjacent with v.

Cartesian product graphs can be recognized efficiently, in time O(m log n) for a graph with m edges and n vertices (Aurenhammer et al., 1992). The operation is essentially commutative, as the graphs G ◻ H and H ◻ G are isomorphic. The operation is also associative, as the graphs (F ◻ G) ◻ H and F ◻ (G ◻ H) are isomorphic.

The notation G × H is occasionally also used for Cartesian products of graphs, but more commonly refers to the tensor product of graphs. The square symbol is the more common and unambiguous notation for the Cartesian product of graphs. It shows visually the four edges resulting from the Cartesian product of two edges.

It should be noted that the cartesian product is not a product in the category of graphs, but only a tensor product (in a category theoretical sense).

Contents 1 Examples 2 Properties 3 References 4 External links

[edit] Examples

• The cartesian product of two edges is a cycle on four vertices: K₂ ◻ K₂ = C₄.
• The cartesian product of two paths is a grid graph.
• The cartesian product of two hypercube graphs is another hypercube: Q_i ◻ Q_j = Q_i+j. More generally the cartesian product of two median graphs is another median graph.
• The graph of vertices and edges of an n-prism is the cartesian product graph K₂ ◻ C_n.
• The Rook's graph is the cartesian product of two complete graphs.

[edit] Properties

If a connected graph is a cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs (Sabidussi 1960; Vizing 1963). However, Imrich and Klavžar (2000) describe a disconnected graph that can be expressed in two different ways as a cartesian product of prime graphs:

(K₁ + K₂ + K₂²) ◻ (K₁ + K₂³) = (K₁ + K₂² + K₂⁴) ◻ (K₁ + K₂),

where the plus sign denotes disjoint union and the superscripts denote exponentiation over cartesian products.

A cartesian product is vertex-transitive if and only if each of its factors is (Imrich and Klavžar, Theorem 4.19).

The chromatic number of the cartesian product satisfies the equation

χ(G ◻ H) = max {χ(G), χ(H)}

(Sabidussi 1957). The Hedetniemi conjecture states a related equality for the tensor product of graphs. The independence number of a cartesian product is not so easily calculated, but as Vizing (1963) showed it satisfies the inequalities

α(G)α(H) + min{|V(G)|-α(G),|V(H)-α(H)} ≤ α(G ◻ H) ≤ min{α(G) |V(H)|, α(H) |V(G)|}.

The Vizing conjecture states that the domination number of a cartesian product satisfies the inequality

γ(G ◻ H) ≥ γ(G)γ(H).

[edit] References

• Aurenhammer, F.; Hagauer, J.; Imrich, W. (1992). "Cartesian graph factorization at logarithmic cost per edge". Comput. Complexity 2: 331–349. doi:10.1007/BF01200428. MR1215316. 

• Imrich, Wilfried; Klavžar, Sandi (2000). Product Graphs: Structure and Recognition. Wiley. ISBN 0-471-37039-8. 

• Sabidussi, G. (1957). "Graphs with given group and given graph-theoretical properties". Canad. J. Math. 9: 515–525. MR0094810. 

• Sabidussi, G. (1960). "Graph multiplication". Math. Z. 72: 446–457. doi:10.1007/BF01162967. MR0209177. 

• Vizing, V. G. (1963). "The Cartesian product of graphs" (In Russian). Vycisl. Sistemy 9: 30–43. MR0209178. 

[edit] External links

• Eric W. Weisstein, Graph Cartesian Product at MathWorld.