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Cycle Decompositions in 3-Uniform Hypergraphs

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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.

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We thank Felix Joos and Allan Lo for helpful discussions and suggestions, and the second author thanks Vincent Pfenninger for useful conversations.

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Correspondence to Simón Piga.

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The research leading to these results was supported by the Czech Science Foundation, Grant Number GA19-08740 S with institutional support RVO: 67985807 (N. Sanhueza-Matamala) while the second author was affiliated with the Institute of Computer Sciences of the Czech Academy of Sciences; and by ANID/CONICYT Acuerdo Bilateral DAAD/62170017 through a Ph.D. Scholarship (S. Piga).

Appendix A: Proof of Lemma 7.3

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

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

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

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

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

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

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,

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

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. \(\square \)

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Piga, S., Sanhueza-Matamala, N. Cycle Decompositions in 3-Uniform Hypergraphs. Combinatorica 43, 1–36 (2023).

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