Totally odd immersions of complete graphs in graph products

TL;DR

This work extends the study of immersions to the stronger notion of totally odd immersions via the parameter $toi(G)$, and investigates how $toi$ behaves under the four standard graph products. The authors prove lower bounds for $toi$ under direct and Cartesian products by constructing disjoint families of odd-path connections between suitably chosen terminals, embedding large complete graphs as totally odd strong immersions in product graphs. They show, in particular, that $toi(G\times H) \ge toi(K_t\times K_s)$ when $toi(G)=t$ and $toi(H)=s$, and develop dense constructions such as $toi(K_{2t}\times K_s)\ge ts$. They also establish that for Cartesian products, $toi(G\square H)\ge toi(K_t\square K_s)$ and provide explicit bounds that are tight in several regimes, ultimately arguing that no minimal counterexample to the immersion analogue of the Odd Hadwiger Conjecture can arise from these products. The results contribute evidence toward a unified product-arity theory for totally odd immersions and clarify the role of graph products in potential counterexamples.

Abstract

For a graph $G$, let $im(G)$ denote the maximum integer $t$ such that $G$ contains $K_t$ as an immersion. A recent paper of Collins, Heenehan, and McDonald (2023) studied the behaviour of this parameter under graph products, asking how large can $im(G\ast H)$ be in terms of $im(G)$ and $im(H)$, when $\ast$ is one of the four standard graph products. We consider a similar question for the parameter $toi(G)$ which denotes the maximum integer $t$ such that $G$ contains $K_t$ as a totally odd immersion. As an application, we obtain that no minimum counterexample to the immersion-analogue of the Odd Hadwiger Conjecture can be obtained from the Cartesian, direct (tensor), lexicographic or strong product of graphs.

Figures (7)

• Figure 1: Let us suppose that the odd path of the totally odd strong immersion of $K_r$ in $K_t\times K_s$ that joins $(x_i,y_j)$ and $(x_{i'},y_j)$ is given by the sequence of vertices $(x_i,y_j)-(x_{i_1},y_{j_1})- (x_{i_2},y_{j_2}) -(x_{i'},y_{j})$. The illustration shows the result of the union of the paths $M^*_{(i,j)-({i_1},{j_1})}$ in green, $M^*_{({i_1},{j_1})-({i_2},{j_2})}$ in red and $M^*_{({i_2},{j_2})-(i',j)}$ in dark red, which corresponds to the odd path joining $(u_i,v_j)$ and $(u_{i'},v_j)$ in the totally odd strong immersion of $K_r$ in $G\times H$.
• Figure 2:
• Figure 8: When $i=1$ and $j\in \{1,2,\dots s\}$
• Figure 9: When $i=t$ and $j\in \{1,2,\dots s\}$
• Figure 11: Case when $(x_1,y_3)$ is the fifth terminal. The red path is $(x_1,y_2)-(x_2,y_2)-(x_2,y_3)-(x_2,y_1)$ and the green path is $(x_1,y_2)-(x_3,y_2)-(x_3,y_3)-(x_3,y_1)$.

Theorems & Definitions (25)

• Conjecture 1.1: Abu-Khzam and Langston AbuLangston
• Conjecture 1.3: Jiménez, Quiroz and Thraves Caro JimenezQuirozThraves
• Theorem 1.4
• Corollary 1.5
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Theorem 2.3
• proof