Minors and Strong Products

Abstract

ANDREȊ KOTLOVLet G H and G2H denote, respectively, the strong and Cartesian products of graphs G and H . (We recall that K 2 K 2 is the complete graph K 4 on four vertices, while K 2 2K 2 is a four-cycle C 4 .) Using a simple construction, we show that, for every bipartite G, the graph G K 2 is a minor of the graph G2C 4 . In particular, the d-cube Q d has a complete minor on 2 (d+1)/2 vertices if d is odd, and on 3 · 2 (d−2)/2 vertices if d is even. We do not know whether such a complete minor of Q d is largest possible.

A note on graph minors and strong products

“…In particular, we show that, for any simple connected graph G, the graph G K 2 is a minor of the graph G Q r by a construction method, where Q r is an r-cube and r = χ (G). This generalizes an earlier result of Kotlov [2].…”

“…is sufficiently dense. For example, for any bipartite graph G which is sufficiently dense, G K 2 G K 3 (see [2]). If G = K 3 ,…”

Hadwiger Number and the Cartesian Product of Graphs

“…It was shown in [2] [10]. He showed that for every bipartite graph G, the strong product ( [7]) G ⊠ K 2 is a minor of G ✷ C 4 .…”

Clique immersion in graph products

“…Using Remark 1 and the fact that ∆(P 2 6 ) = 4 we can conclude im(P 2 6 ) = 5. As mentioned in the introduction, Kotolov [14] and Chandran and Sivadasan…”

“…For the Cartesian product of a path on n vertices with itself d times, denoted P d n , we show im(P d n ) = 2d + 1. We compare the results for hypercubes, Hamming graphs, and P d n to results by Kotlov [14] and Chandran and Sivadasan [3] for graph minors. In addition, we show we can do better than the bound of Theorem 8 by proving im(G P n ) ≥ t + 2 when im(G) = t.…”