Cycle Decompositions in 3-Uniform Hypergraphs

Abstract

We show that 3-graphs on n vertices whose minimum codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing that our bounds are best possible up to a o(1) term. All together, our results answer negatively some recent questions of Glock, Joos, Kühn, and Osthus.

Appendix A: Proof of Lemma 7.3

Proof The proof proceeds in three steps. First, we find $H_p\subseteq H$ by including each edge with probability p, and in the remainder $H_0 = H \setminus H_p$ we find an almost perfect $C_{\ell }$-packing ${\mathcal {C}}_0$, let $L_0 = H_0 \setminus E({\mathcal {C}}_0)$ be the leftover edges. Secondly, we correct the leftover $L_0$ in the vertices incident with $\Omega (n^2)$ many edges of $L_0$ by constructing cycles with the help of the edges in $H_p$. This provides us with a new cycle packing ${\mathcal {C}}_1\subseteq L_0 \cup H_p$ whose new leftover $L_1 = H_0 \setminus E({\mathcal {C}}_0 \cup {\mathcal {C}}_1)$ satisfies $\Delta _1(L_1) = o(n^2)$. Finally, we correct the new leftover $L_1$ in a similar way, fixing the pairs incident to $\Omega (n)$ edges in $L_1$. We get a cycle packing ${\mathcal {C}}_2\subseteq L_1 \cup H_p$, and ${\mathcal {C}}_0 \cup {\mathcal {C}}_1 \cup {\mathcal {C}}_2$ will be the desired cycle packing.

Step 1: Random slice and aproximate decomposition. Note that $\delta ^{(3)}_2(H) \ge 3 \varepsilon n$. Now let $p = \gamma / 4$, and let $H_p\subseteq H$ be obtained from H by including each edge independently with probability p. Using concentration inequalities (e.g. Theorem 5.4) we see that with non-zero probability

$$\begin{aligned} \Delta _2(H_p) \le 2 p n, \text { and } \delta ^{(3)}_2(H_p) \ge 2 \varepsilon p n. \end{aligned}$$

(A.1)

hold simultaneously for $H_p$. From now on we suppose $H_p$ is fixed and satisfies (A.1).

Let $H_0 = H \setminus H_p$. In $H_0$, construct a $C_{\ell }$-packing by removing edge-disjoint cycles, one by one, until no longer possible. We get a $C_\ell$-packing ${\mathcal {C}}_0$ in $H_0$, let $F_0 = E({\mathcal {C}}_0)$. By Erdős’ Theorem [18, Theorem 1] there exists $c > 0$ such that $L_0 = H_0 \setminus F_0$ has at most $n^{3 - 3c}$ edges.

Step 2: Eliminating bad vertices. Let $B_0 = \{ v \in V: \deg _{L_0}(v) \ge n^{2 - 2c} \}$. Since $|L_0| \le n^{3 - 3c}$, by double-counting we have $|B_0| \le 3 n^{1 - c}$.

For each $b \in B_0$, let $G_b$ be the subgraph of $L_0(b)$ obtained after removing the vertices of $B_0$. Note that $L_0(b) - G_0$ has at most $|B_0|n \le 3n^{2-c}$ edges. Now, let ${\mathcal {P}}_b$ be a maximal edge-disjoint collection of paths of length 3 in $G_b$. Since every graph on n vertices with at least $n+1$ edges contains a path of length 3, then $G_b - E({\mathcal {P}}_b)$ has at most n edges. All together, we deduce that the number of edges in $L_0(b) - E({\mathcal {P}}_b)$ satisfies

$$\begin{aligned} |L_0(b)| - |E({\mathcal {P}}_b)| \le 3n^{2-c} + n \le 4 n^{2-c}. \end{aligned}$$

(A.2)

Since $G_b$ contains at most $n^2$ edges, we certainly have $|{\mathcal {P}}_b| \le n^2$. Let ${\mathcal {P}}_b$ be a collection of tight paths on five vertices obtained by replacing each $v_0 v_1 v_2 v_3$ in ${\mathcal {P}}_b$ with the tight path $v_0 v_1 b v_2 v_3$ in $L_0$. Note that any two distinct $P_1, P_2 \in {\mathcal {P}}_b$ are edge-disjoint, and for two distinct $b, b' \in B_0$, and $P \in {\mathcal {P}}_b$, $P' \in {\mathcal {P}}_{b'}$, since $b' \notin V(G_b)$ we have $P, P'$ are edge-disjoint. Thus the union ${\mathcal {P}}= \bigcup _{b \in B_0} {\mathcal {P}}_b$ is an edge-disjoint collection of tight paths on 5 vertices.

Select $\gamma ', \mu ', \varepsilon '$ such that $1/n \ll \gamma ' \ll \mu ' \ll \varepsilon ' \ll \gamma , \varepsilon , 1/\ell$. We wish to apply Lemma 7.1 to extend ${\mathcal {P}}$ into cycles. We claim ${\mathcal {P}}$ is $\gamma '$-sparse. Let $S \in \left( {\begin{array}{c}V(H)\\ 2\end{array}}\right)$. Since $|{\mathcal {P}}| \le |B_0| n^2 \le 3 n^{3 - c} \le \gamma ' n^{3}$, certainly ${\mathcal {P}}$ contains at most $|{\mathcal {P}}| \le \gamma ' n^{3}$ paths of type 0 for S. Now, note that for each $b \in B_0$, $P \in {\mathcal {P}}_b$ can have at most 2n paths of type 1 for S, thus ${\mathcal {P}}$ has at most $|B_0|2n \le 6n^{2-c} \le \gamma ' n^{2}$ paths of type 1 for S. Analogously, for each $b \in B_0$, $P \in {\mathcal {P}}_b$ can have at most 1 path of type 2 for S, thus ${\mathcal {P}}$ has at most $|B_0| \le 3n^{1-c} \le \gamma ' n$ paths of type 2 for S. Thus ${\mathcal {P}}$ is $\gamma '$-sparse.

Recall that $L_0$ is edge-disjoint with $H_p$. Inequalities (A.1) together with $p = \gamma /4$ and $\varepsilon ' \ll \gamma , 1/\ell$, show that we can use Corollary 5.3 (with $U = V(H)$) and deduce that for each $P \in {\mathcal {P}}$, there exists at least $\varepsilon ' n^{\ell - 5}$ copies of $C_{\ell }$ in $L_0 \cup H_p$ that extend $P_i$ using extra edges of $H_p$ only.

We apply Lemma 7.1 with $\varepsilon ', \mu ', \gamma ', \ell , 5, L_0, H_p, {\mathcal {P}}$ playing the rôle of $\varepsilon , \mu , \gamma , \ell , \ell ', H_1, H_2, {\mathcal {P}}$ respectively, to obtain a $C_{\ell }$-decomposable graph $F_1\subseteq L_0 \cup H_p$ such that $E({\mathcal {P}})\subseteq F_1$ and

$$\begin{aligned} \Delta _2(F_1 \setminus E({\mathcal {P}})) \le \mu 'n. \end{aligned}$$

(A.3)

Since $F_0$, $F_1$ are edge-disjoint, $F_0 \cup F_1$ is $C_{\ell }$-decomposable. Let $L_1 = H_0 \setminus ( F_0 \cup F_1 )$. Observe that, if $v \notin B_0$, then $\deg _{L_1}(v) \le \deg _{L_0}(v) <n^{2-2c}$ by definition. Moreover, if $v \in B_0$, then each edge in $E({\mathcal {P}}_v)$ is in $F_1$, and hence (A.2) implies $\deg _{L_1}(v) \le |L_0(v)| - |E({\mathcal {P}}_v)| \le 4n^{2-c}$. Therefore,

$$\begin{aligned} \Delta _1(L_1) \le 4n^{2-c}. \end{aligned}$$

(A.4)

Step 3: Eliminating bad pairs. Let $f = c/2$ and $B_1 = \{ xy \in \left( {\begin{array}{c}V\\ 2\end{array}}\right) : \deg _{L_1}(xy) \ge n^{1 - f} \}$. From $|L_1| \le |L_0| \le n^{3 - 3c} \le n^{3 - 6f}$ we deduce $|B_1| \le n^{2 - 4f}$. Now consider $B_1$ as the set of edges of a 2-graph in V. Each edge of $B_1$ incident to a vertex x implies that x belongs to at least $n^{1 - f}$ edges in $L_1$, and each of those edges participates in at most two of the edges in $B_1$ incident to x. So we have $\deg _{L_1}(x) \ge \frac{1}{2} n^{1 - f} \deg _{B_1}(x)$. Together with inequality (A.4) we deduce $\Delta (B_1) \le 8 n^{1 - f}$.

A path P on $L_1$ is $B_1$-based if $P =zxyw$ and $xy \in B_1$. Let ${\mathcal {P}}_2$ be a maximal packing of $B_1$-based paths. For all $xy \in B_1$, it holds that $\deg _{L_1}(xy) - \deg _{E({\mathcal {P}}_2)}(xy) \le 1$. Otherwise it would exist distinct $z, w \in N_{L_1 \setminus E({\mathcal {P}}_2)}(xy)$, and then zxyw would a $B_1$-based path not in ${\mathcal {P}}_2$, which contradicts its maximality.

We claim ${\mathcal {P}}_2$ is $\gamma '$-sparse. For each $xy \in B_1$, let ${\mathcal {P}}_{xy}\subseteq {\mathcal {P}}_2$ be the paths whose two interior vertices are precisely xy. Clearly $|{\mathcal {P}}_{xy}| \le n$ and ${\mathcal {P}}_2 = \bigcup _{xy \in B_1} {\mathcal {P}}_{xy}$. Let $e \in \left( {\begin{array}{c}V\\ 2\end{array}}\right)$. Since $|{\mathcal {P}}_2| \le \sum _{xy \in B_1} |{\mathcal {P}}_{xy}| \le n |B_1| \le n^{3 - 4f} \le \gamma ' n^3$, there are at most $\gamma ' n^3$ paths of type 0 for e in ${\mathcal {P}}_2$. Recall that if $P = zxyw$ is a path of type 1 for e, then we have $|e \cap \{z, x, y, w\}| = 1$. If $xy \in B_1$ satisfies $e \cap \{x, y\} = \emptyset$, then at most two paths in ${\mathcal {P}}_{xy}$ can be of type 1 for e and therefore there are at most $2|B_1| \le 2n^{2 - 4f}$ paths of type 1 for e in ${\mathcal {P}}_2$. We estimate the contribution of the pairs $xy \in B_1$ such that $|e \cap \{x,y\}| = 1$. Each such xy contributes at most n paths of type 1 for e in ${\mathcal {P}}_{xy}$. By (A.4), the number of such xy is at most $2 \Delta (B_1) \le 16 n^{1 - f}$, thus the total contribution of those pairs is at most $16 n^{2-f}$. All together, the total number of paths of type 1 for e in ${\mathcal {P}}_2$ is at most $2n^{2 - 4f}+16 n^{2-f} \le \gamma ' n^2$. If $e = \{a,b\}$ then ${\mathcal {P}}_{a,b}$ does not contain any path of type 2 for e, by definition of the path types. Thus the only possible contributions come from the pairs in ${\mathcal {P}}_{a,x}$ and ${\mathcal {P}}_{b,y}$ for some $x, y \in V(H)$; and each one of those sets contains at most 1 path of type 2 for e. Thus the total number of pairs of type 2 for e in ${\mathcal {P}}_2$ is at most $2 \Delta (B_1) \le 16 n^{1 - f} \le \gamma ' n$. Thus ${\mathcal {P}}_2$ is $\gamma '$-sparse.

Let $H'_p = H_p \setminus ( F_0 \cup F_1 )$. (A.1) and (A.3), together with $\mu ' \ll \varepsilon ' \ll \gamma , 1/\ell$, allow us to use Corollary 5.3 with $U = V(H)$, thus for each $P \in {\mathcal {P}}_2$, there exist at least $\varepsilon ' n^{\ell - 4}$ copies of $C_{\ell }$ in $L_1 \cup H'_p$ that extend P using extra edges of $H'_p$ only. Apply Lemma 7.1 with the parameters $\varepsilon ', \mu ', \gamma ', \ell , 4, L_1, H'_p, {\mathcal {P}}_2$ playing the roles of $\varepsilon , \mu , \gamma , \ell , \ell ', H_1, H_2, {\mathcal {P}}$ respectively, to obtain a $C_{\ell }$-decomposable $F_2\subseteq L_1 \cup H'_p$ such that $E({\mathcal {P}}_2)\subseteq F_2$ and $\Delta _2(F_2 \setminus E({\mathcal {P}}_2)) \le \mu 'n$.

We claim that $\Delta _2(L_1 \setminus F_2) \le n^{1 - f}$. Indeed, if $xy \in B_1$, $\deg _{L_1 \setminus F_2}(xy) \le \deg _{L_1}(xy) \le n^{1-f}$ follows by definition, otherwise, $E({\mathcal {P}}_2) \subseteq F_2$ implies $\deg _{L_1 \setminus F_2}(xy) \le \deg _{L_1}(xy) - \deg _{F_2}(xy) \le 1$. Since $F_2$ and $F_0 \cup F_1$ are edge-disjoint, $F = F_0 \cup F_1 \cup F_2$ is a $C_{\ell }$-decomposable subgraph of H. We claim $L = H \setminus F$ satisfies $\Delta _2(L) \le \gamma n$. Indeed, an edge not covered by F is either in $H_p$ or in $L_1 \setminus F_2$. Thus we have $\Delta _2(L) \le \Delta _2(H_p) + \Delta _2(L_1 \setminus F_2) \le 2pn + n^{1-f} \le \gamma n$, as required. $\square$