Related papers: Counterexamples to Clique Immersion Conjecture for…

Let $G,H$ be graphs and $G*H$ represent a particular graph product of $G$ and $H$. We define $im(G)$ to be the largest $t$ such that $G$ has a $K_t$-immersion and ask: given $im(G)=t$ and $im(H)=r$, how large is $im(G*H)$? Best possible…

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…

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, and all the paths…

An immersion of a graph $H$ in a graph $G$ is a minimal subgraph $I$ of $G$ for which there is an injection ${{\rm i}} \colon V(H) \to V(I)$ and a set of edge-disjoint paths $\{P_e: e \in E(H)\}$ in $I$ such that the end vertices of…

We prove that if $L(G)$ immerses $K_t$ then $L(mG)$ immerses $K_{mt}$, where $mG$ is the graph obtained from $G$ by replacing each edge in $G$ with a parallel edge of multiplicity $m$. This implies that when $G$ is a simple graph, $L(mG)$…

In \cite{nr-1996} Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that $i(G\times H) \ge i(G)i(H)$ where $i(G)$ is the independent domination number of $G$ and…

The immersion-analogue of Hadwiger's Conjecture states that every graph $G$ contains an immersion of $K_{\chi(G)}$. This conjecture has been recently strengthened in the following way: every graph $G$ contains a totally odd immersion of…

The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{\chi (G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $\alpha$ contains…

The Lescure-Meyniel conjecture is the analogue of Hadwiger's conjecture for the immersion order. It states that every graph $G$ contains the complete graph $K_{\chi(G)}$ as an immersion, and like its minor-order counterpart it is open even…

The immersion number of a graph $G$, denoted im$(G)$, is the largest $t$ such that $G$ has a $K_t$-immersion. In this note we are interested in determining the immersion number of the $m$-Mycielskian of $G$, denoted $\mu_m(G)$. Given the…

A graph is said to contain $K_k$ (a clique of size $k$) as a weak immersion if it has $k$ vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger's conjecture:…

We consider the problem of how much edge connectivity is necessary to force a graph G to contain a fixed graph H as an immersion. We show that if the maximum degree in H is D, then all the examples of D-edge connected graphs which do not…

Let $G$ be a graph and $r\in\mathbb{N}$. The matching Kneser graph $\textsf{KG}(G, rK_2)$ is a graph whose vertex set is the set of $r$-matchings in $G$ and two vertices are adjacent if their corresponding matchings are edge-disjoint. In…

The analogue of Hadwiger's conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{\chi(G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph…

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from…

Let $h(G)$ denote the largest $t$ such that $G$ contains $K_t$ as a minor and $\chi(G)$ be the chromatic number of $G$ respectively. In 1943, Hadwiger conjectured that $h(G) \geq \chi(G)$ for any graph $G$. In this paper, we prove that…

Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Gy\H{o}ri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Tur\'{a}n graph on $n$ vertices and $r - 1$ parts,…

In this article we consider the relationship between vertex coloring and the immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston in 2003, which says that the complete graph with $t$ vertices can be immersed in any…

Building on recent work of Dvo\v{r}\'ak and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t + 4$ contains $K_t$ as an immersion.…

We give a Hilton-Milner Theorem for the $r$-independent sets in the graph that is the union of copies of $K_k$. That is, we determine the maximum intersecting families of $r$-independent sets in this graph, subject to the condition that the…