arXiv subjects
Publications and source records attributed to Henry Echeverría.
We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\ell\geq χ(G)+1$, the reconfiguration graph of proper $\ell$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open case in the classification of recolorable $(F_1,F_2)$-free graphs when $F_1$ and $F_2$ have at most four vertices.
Given a graph $G$ and an integer $\ell>χ(G)$, the reconfiguration graph of the $\ell$-colorings of $G$ has as its vertices as the proper $\ell$-colorings of $G$, with an edge between two colorings whenever they differ on exactly one vertex. We say that $G$ is \emph{recolorable} if this reconfiguration graph is connected for every $\ell>χ(G)$. Belavadi and Cameron determined which $(F_1,F_2)$-free graphs are recolorable whenever $F_1$ and $F_2$ are graphs on at most four vertices, with the single exception of $(2K_2,K_4)$-free graphs. Gaspers and Huang showed such graphs are $4$-colorable. The $3$-colorable case within this class has also been resolved, leaving the open question of whether every $(2K_2,K_4)$-free graph with chromatic number $4$ is recolorable. In this paper, we provide evidence toward an affirmative answer by establishing recolorability for three subclasses: $(2K_2,K_4,C_5)$-free graphs, $(2K_2,K_4,H_a,H_b)$-free graphs for any distinct $a,b\in \{2,3,4\}$, and $(2K_2,K_4,H_4)$-free graphs containing an induced $W_5$, where $H_i$ denotes the unique $2K_2$-free graph obtained from a $W_5$ by keeping exactly $i$ edges from the universal vertex to the cycle.
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.
We study the chromatic number of graphs that exclude a clique as a strong odd immersion and have independence number two. Given a graph $G$ and $t\in\mathbb{Z}^+$, we prove that if $α(G)\leq 2$ and $G$ has no strong odd $K_t$-immersion, then $χ(G)\leq \lceil \frac{3(t-1)}{2}\rceil$.
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.
As evidence for the Odd Hadwiger Conjecture, Simonyi and Zsbán (2010) showed that every Kneser graph $G$ with large enough order (compared to $χ(G)$) contains a totally odd subdivision of $K_{χ(G)}$. A recent result of Steiner (2024), shows that every Schriver graph, and thus every Kneser graph, satisfies the Odd Hadwiger Conjecture, that is, it contains $K_{χ(G)}$ as an odd minor. We strengthen these results for Kneser graphs in two ways. We show that for every $t\ge 8$, there are $t$-chromatic Kneser graphs that contain arbitrarily large complete totally odd subdivisions (and thus, odd minors). We also show that every Kneser graph contains a totally odd subdivision of $K_{χ(G)}$. Kneser graphs are the prime example of graphs having chromatic number equal to its topological lower bounds. Motivated by our main results, we also study totally odd immersions on graphs with this property, proving, in particular, that if the chromatic number of $G$ is equal to any of its topological lower bounds, then $G$ contains a totally odd immersion of $K_{\lfloor χ(G)/2 \rfloor +1}$. This gives evidence for the immersion-analogue of the Odd Hadwiger Conjecture.