The asymptotic version of the Erdős-Sós conjecture and beyond

Abstract

Klimošová, Piguet, and Rozhoň conjectured that any graph with minimum degree $k/2$ and sufficiently many vertices of degree $k$ should contain all trees with $k$ edges. We prove an asymptotic version of this conjecture for dense host graphs. We obtain interesting corollaries: the first is an asymptotic version of the Erdős--Sós conjecture for dense host graphs, which works without any bounded-degree restriction on the guest trees. Secondly, by leveraging recent results by Pokrovsky, we can translate our results to sparse host graphs in the case of bounded-degree guest trees.

Figures (7)

• Figure 1: A schematic view of an $\ell$-fine partition of a tree with $45$ vertices rooted at $r$ into $W_A, W_B, \mathcal{F}_A, \mathcal{F}_B$, satisfying \ref{['item:vertex partition']}--\ref{['item:fp-distance']}.
• Figure 2: A view of a $(G, S, M)$ Gallai--Edmonds triple. Note that however there might be more edges, here we just draw the matching $M_S$ between $S$ and components of $G-S$ by dashed edges. We also illustrate the notion of a factor-critical component by completing $M_S$ with a matching (full edges) in each non-singleton component of $\mathcal{K}^*-V(M_S)$.
• Figure 3: An alternating path $P_u$ in bold red edges starting at $u$ in a $(G, S, M)$ Gallai--Edmonds triple structure. The support of the fractional matching $\mu$ is illustrated in dashed edges. For any vertex $v$ in $V(P_u)\cap S$, the set $N_\mu(v)$ (shown in blue) consists of the neighbours of $v$ connected by support edges of $\mu$.
• Figure 4: Situations in the proof of \ref{['lemma:separatinglemma-1']}. In the second figure $\textcolor{red}{\tilde{\mu}'}+\textcolor{blue}{\tilde{\sigma}'}$ covers $x$ and $y$ as in the first figure, but additionally it covers also half of $d$. Hence $\deg_w(c, \textcolor{red}{\tilde{\mu}'}+\textcolor{blue}{\tilde{\sigma}'})>\deg_w(c, \textcolor{red}{\tilde{\mu}}+\textcolor{blue}{\tilde{\sigma}})$. Therefore, $(\textcolor{red}{\tilde{\mu}}, \textcolor{blue}{\tilde{\sigma}})$ was not an optimal GE pair, as assumed.
• Figure 5: The new GE pair created in the right picture saturates the neighbourhood of $c$ by the same amount as the one on the left. As $xy$ is an edge the GE pair $(\textcolor{blue}{\tilde{\sigma}}, \textcolor{red}{\mu_\delta})$ cannot be optimal, and thus neither is $(\textcolor{blue}{\tilde{\sigma}}, \textcolor{red}{\tilde{\mu}})$, contradicting the assumption of \ref{['lemma:separatinglemma-2']}.

Theorems & Definitions (115)

• theorem 1: Main result
• corollary 1
• corollary 2
• corollary 3
• proof : Proof of \ref{['cor:dense-approx E-S']}
• theorem 2
• theorem 3
• proof : Proof of \ref{['cor:approDense-MinMax-sparse']}
• proof : Proof of \ref{['corollary:ramsey']}
• lemma 1: Tree-Coating Lemma