Odd Hadwiger number and graph products

Abstract

The Odd Hadwiger number of a graph $G$ is the largest integer $r$ such that $G$ has a clique of size $r$ as an odd minor. In this paper, we investigate how large is the Odd Hadwiger number of the product of two graphs, when considering any of the four standard graph products: Cartesian, direct, lexicographic, strong. We provide an optimal lower bound in the cases of the strong and lexicographic products.

Figures (5)

• Figure 1: The Cartesian product of an odd expansions of $K_3$ and an odd expansion of $K_2$. The subgraphs $S_i\Square T_j$ are depicted with black edges, with bold black edges representing the spanning trees $\mathscr{T}_{ij}$. Moreover, the red edges are the corresponding monochromatic edges in the odd expansions. The colour 1 from $c_1$ is depicted as black and the colour 2 as white.
• Figure 2: Trees $Z_1, Z_2, \ldots, Z_{s+4}$ and $C$ from the Theorem \ref{['Theorem:CartesianComplete']} in the case that $t=6$. The colour 1 is represent by black and the colour 2 by red. Note that $Z_1, \ldots, Z_6$ are vertices, the vertex $(u_1,v_6)$ is not used in any tree in $\mathbf{Z}$, and $Z_7, \ldots, Z_{s+4}$ are stars with 5 leaves.
• Figure 3: Case $t=6$. Recall that $Z_8$ is defined as $(u_6,v_1)-(u_4,v_3)-(u_6,v_2)$.
• Figure 4: Case $t=7$. Tree $Z_9$ is obtained from the row $t+2$ odd in Table \ref{['table:trees1']}.
• Figure 6: The figure illustrates a $K_{12}$ as an odd minor of the graph $K_6\times K_6$ from Theorem \ref{['thm:complete_direct']}. Also, the tress and the 2-colouring of it are showed, the colour 1 is red and the colour 2 is black. There are two singletons trees represented by $(u_1,v_1)$, and $(u_3,v_3)$. White vertices are not used in the construction of the odd minor.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof