# Prague Dimension of Random Graphs

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## Abstract

The Prague dimension of graphs was introduced by Nešetřil, Pultr and Rödl in the 1970s. Proving a conjecture of Füredi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order $$n/\log n$$ for constant edge-probabilities. The main new proof ingredient is a Pippenger–Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size $$O(\log n)$$.

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## Notes

1. The decision problem of whether $${\text {dim}}_{\textrm{P}}(G) \le k$$ holds is also known to be NP-complete for $$k \ge 3$$, see [33].

2. As usual, we say that an event holds whp (with high probability) if it holds with probability tending to 1 as $$n\rightarrow \infty$$.

3. Many deterministic approaches such as [22, 37] first efficiently color most of the edges of $${\mathcal {H}}$$ using $${(1+\delta /2)\Delta ({\mathcal {H}})}$$ colors, say, so that the remaining uncolored ‘last few edges’ yield a hypergraph with maximum degree at most $${\epsilon \Delta ({\mathcal {H}})}$$, say. By choosing the constant $${\epsilon =\epsilon (r,\delta )>0}$$ sufficiently small, these ‘last few edges’ can then trivially be colored using $${r\cdot \epsilon \Delta ({\mathcal {H}})} \le {\delta /2 \cdot \Delta ({\mathcal {H}})}$$ additional colors, which clearly becomes harder to implement when $$r=r(n) \rightarrow \infty$$ (as now the dependence of $$\epsilon$$ on r matters).

4. Heuristically, the form of the upper bound (11) can also be motivated as follows: (7) and $$G_i \approx G_{n,{p_i}}$$ loosely suggest $$\text {cc}'(G_{n,p}) \le \sum _{0 \le i \le I}\text {cc}'(G_{n,{p_i}})$$, which together with (6) and $$\text {cc}'(G_{n,{p_I}}) \le 2\Delta (G_{n,{p_I}}) = O(np_I)$$ makes the first inequality in (11) a natural target bound (the second inequality is more technical, and follows by integral comparison; see Sect. 2.2.1).

5. To see the claimed bounds in (17), note that $$1/p_i \le 1/p_I \le k^{2\tau }/p \le n^{\sigma /\tau +o(1)}$$ and $$p_i^{k_i-1} \ge p_i^{\sigma \log _{1/p_i} n} = n^{-\sigma }$$.

6. In fact, inequality (29) holds with probability at least $$1-n^{-\omega (1)}$$, since each of the $$O(I \cdot n) = n^{O(1)}$$ many events considered fails with probability at most $$n^{-\omega (1)}$$. A similar remark applies to inequality (34) in Sect. 2.2.2.

7. We consider the auxiliary hypergraph $${\mathcal {H}}$$, where the vertices correspond to the edges of $$G_{n,p}$$ and the edges correspond to the edge-sets of the cliques $$K_s$$ of $$G_{n,p}$$. The technical conditions of [23, Theorem 7.1] required for mimicking [23, Section 7] can then be verified using (careful applications of) standard tail bounds such as Lemma 11 and [46, Theorems 30 and 32].

8. For the same auxiliary hypergraph $${\mathcal {H}}$$ as considered before, the required technical conditions of [11, Theorem 1.2] with $$\Delta \approx \left( {\begin{array}{c}n-2\\ s-2\end{array}}\right) p^{\left( {\begin{array}{c}s\\ 2\end{array}}\right) -1} \ge \Omega \bigl ((\log n)^{\omega (1)}\bigr )$$ and $$\log e({\mathcal {H}}) \le s \log n \ll \Delta ^{\Theta (1)}$$ can be verified using Lemma 11 and [40, Theorem 1].

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## Acknowledgements

We would like to thank Annika Heckel for valuable discussions about Problem 1. We are also grateful to the anonymous referees for useful suggestions concerning the presentation.

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Correspondence to Lutz Warnke.

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Supported by NSF grant DMS-1703516, NSF CAREER grant DMS-2225631, and a Sloan Research Fellowship.

## Appendices

### Proof of Lemma 20

Writing $${\mathcal {S}}$$ for the event that the largest clique of $$G_{n,p}$$ has size at most $$s=\lceil 2\log _{1/p} n\rceil$$, it well-known that $${\mathcal {S}}$$ holds whp (by a straightforward first moment argument). Writing $${\mathcal {E}}$$ for the event that $$G_{n,p}$$ contains $$(1\pm \epsilon )\left( {\begin{array}{c}n\\ 2\end{array}}\right) p$$ edges for $$\epsilon :=n^{-1/2}$$, say, it is easy to see that $${\mathcal {E}}$$ holds whp (using Chebychev’s inequality). Furthermore, recalling $$\varphi (p)={(1-p)\log (1-p)/(p\log p)}$$, the probability that $$G_{n,p}$$ equals any fixed spanning subgraph $$G \subseteq K_n$$ with $$e(G) = (1\pm \epsilon )\left( {\begin{array}{c}n\\ 2\end{array}}\right) p$$ edges is routinely seen to be at most

\begin{aligned} \begin{aligned} \Pi&:= \max _{m \in (1\pm \epsilon )\left( {\begin{array}{c}n\\ 2\end{array}}\right) p} p^m (1-p)^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) -m} \\&= \max _{m \in (1\pm \epsilon )\left( {\begin{array}{c}n\\ 2\end{array}}\right) p} \exp \biggl ( -\genfrac(){0.0pt}1{n}{2}p \biggl [\frac{m}{\genfrac(){0.0pt}1{n}{2}p} + \biggl (1-\frac{m}{\genfrac(){0.0pt}1{n}{2}}\biggr )\frac{\log (1-p)}{p\log p}\biggr ] \! \cdot \! \log (1/p)\biggr )\\&\le \exp \biggl ( -\genfrac(){0.0pt}1{n}{2}p \biggl [1-\epsilon + \biggl (1-\frac{\epsilon p}{1-p}\biggr )\varphi (p)\biggr ] \! \cdot \! \log (1/p)\biggr )\\&\le \exp \Bigl (-(1-o(1)) \! \cdot \! \genfrac(){0.0pt}1{n}{2}p \bigl (1+\varphi (p)\bigr ) \! \cdot \! \log (1/p)\Bigr ), \end{aligned} \end{aligned}
(77)

where we used $$\varphi (p) \ge 0$$ as well as $$\epsilon =o(1)$$ and $$\epsilon p/(1-p)=o(1)$$ for the last inequality.

For the clique covering number $$\text {cc}(G_{n,p})$$, the crux is that there are at most

\begin{aligned} \left( {\begin{array}{c}n+s\\ s\end{array}}\right) ^T \, \le \, o(n^{sT}) \end{aligned}

many collections $$\{C_1, \ldots , C_t\}$$ with $$t \le T$$ that are a clique covering for some graph $$G \subseteq K_n$$ with largest clique of size at most s. Hence, since each clique covering uniquely determines the entire edge-set and thus the underlying spanning subgraph $$G \subseteq K_n$$, it follows by a union bound argument that

\begin{aligned} {{\mathbb {P}}{}}(\text {cc}(G_{n,p}) \le T) \; \le \; {{\mathbb {P}}{}}(\lnot {\mathcal {S}}\text { or } \lnot {\mathcal {E}}) \, + \, o(n^{sT}) \cdot \Pi . \end{aligned}
(78)

Note that $${{\mathbb {P}}{}}(\lnot {\mathcal {S}}\text { or } \lnot {\mathcal {E}})=o(1)$$ and $$s\log n \sim \left( {\begin{array}{c}s\\ 2\end{array}}\right) \cdot \log (1/p)$$. In view of inequality (77), for any $$\epsilon \in (0,1)$$ it follows that (78) is at most o(1) when $$T \le (1-\epsilon ) \cdot (1+\varphi (p)) \left( {\begin{array}{c}n\\ 2\end{array}}\right) p/\left( {\begin{array}{c}s\\ 2\end{array}}\right)$$, establishing (75).

Turning to the thickness $$\text {cc}_{\Delta }(G_{n,p})$$, we associate each clique covering $${\mathcal {C}}$$ of some graph $$G \subseteq K_n$$ with an auxiliary bipartite graph $${\mathcal {B}}$$ on vertex-set $${[n] \cup {\mathcal {C}}}$$, where $${v \in [n]}$$ and $${C_i \in {\mathcal {C}}}$$ are connected by an edge whenever $${v \in V(C_i)}$$. If the thickness of $${\mathcal {C}}$$ is at most T, then in $${\mathcal {B}}$$ the degree of each $$v \in [n]$$ is at most $$\lfloor T\rfloor$$, which also gives $${|{\mathcal {C}}| \le n \lfloor T\rfloor }$$. Since the structure of the auxiliary bipartite graph $${\mathcal {B}}$$ uniquely determines $${\mathcal {C}}$$ (as the neighbors of $$C_i$$ in $${\mathcal {B}}$$ determine the clique vertex-set $$V(C_i)$$), it follows that there are at most

\begin{aligned} \left( {\begin{array}{c}n\lfloor T\rfloor + \lfloor T\rfloor \\ \lfloor T\rfloor \end{array}}\right) ^n \, \le \, O\bigl ((6n)^{nT}\bigr ) \end{aligned}

many collections $${\mathcal {C}}$$ with thickness at most T that are a clique covering of some graph $$G \subseteq K_n$$. Since each such $${\mathcal {C}}$$ uniquely determines the underlying spanning subgraph $$G \subseteq K_n$$, we obtain similarly to (78) that

\begin{aligned} {{\mathbb {P}}{}}(\text {cc}_{\Delta }(G_{n,p}) \le T) \; \le \; {{\mathbb {P}}{}}(\lnot {\mathcal {E}}) \, + \, O\bigl ((6n)^{nT}\bigr ) \cdot \Pi . \end{aligned}
(79)

Note that $${{\mathbb {P}}{}}(\lnot {\mathcal {E}})=o(1)$$ and $$n \log (6n) \sim \left( {\begin{array}{c}n\\ 2\end{array}}\right) \log (1/p) \cdot (s-1)/n$$. In view of inequality (77), for any $$\epsilon \in (0,1)$$ it follows that (79) is at most o(1) when $$T \le (1-\epsilon ) \cdot (1+\varphi (p)) np/(s-1)$$, completing the proof of (76). $$\square$$

### Proof of Corollary 10

Choosing $$\xi =\xi (\delta ) \in (0,1/16]$$ such that $$(1+\delta )(1+\xi )/(1-4\xi )^2 \le 1+2\delta$$, set $${m_0:= \lfloor (1+\xi )m\rfloor }$$, $${m_1:= \lfloor m_0/(1-4\xi )^2\rfloor }$$, and $${c:= (1+\delta )r m_1/n}$$. Let $${\mathcal {H}}_{i}^*$$ be chosen uniformly at random from all $$\left( {\begin{array}{c}|E({\mathcal {H}})|\\ i\end{array}}\right)$$ subhypergraphs of $${\mathcal {H}}$$ with exactly i edges. Since $${\mathcal {H}}_q$$ conditioned on having exactly i edges has the same distribution as $${\mathcal {H}}_i^*$$, by the law of total probability and monotonicity it follows that

\begin{aligned} \begin{aligned} {{\mathbb {P}}{}}(\chi '({\mathcal {H}}_q) \ge c)&\, \le \, {{\mathbb {P}}{}}(|E({\mathcal {H}}_q)| > m_0) + \sum _{0 \le i \le m_0}{{\mathbb {P}}{}}(\chi '({\mathcal {H}}_i^*) \ge c){{\mathbb {P}}{}}(|E({\mathcal {H}}_q)| = i) \\&\, \le \, n^{-\omega (r)} + {{\mathbb {P}}{}}(\chi '({\mathcal {H}}_{m_0}^*) \ge c), \end{aligned} \end{aligned}
(80)

where we used standard Chernoff bounds (such as [21, Theorem 2.1]) and $${{\mathbb {E}}{}}|E({\mathcal {H}}_q)| = |E({\mathcal {H}})|q = m \ge n^{1+\sigma } \gg r \log n$$. Sequentially choosing the random edges $$e_1, \ldots , e_{m_1} \in E({\mathcal {H}})$$ of $${\mathcal {H}}_{m_1}$$ as defined in Theorem 2, note that $$e_{i+1} \in {E({\mathcal {H}}) {\setminus } \{e_1, \ldots , e_i\}}$$ holds with probability at least $${1-m_1/e({\mathcal {H}}) > 1-4\xi }$$, as $$m_1 < 4 m \le 4\xi e({\mathcal {H}})$$. Since we can equivalently construct the edge-set $$\{f_1, \ldots , f_{m_0}\}$$ of $${\mathcal {H}}^*_{m_0}$$ by sequentially choosing $$f_{i+1} \in {E({\mathcal {H}}) {\setminus } \{f_1, \ldots , f_i\}}$$ uniformly at random, a natural coupling of $${\mathcal {H}}_{m_1}$$ and $${\mathcal {H}}^*_{m_0}$$ thus satisfies

\begin{aligned} \begin{aligned} {{\mathbb {P}}{}}({\mathcal {H}}_{m_0}^* \subseteq {\mathcal {H}}_{m_1}) \, \ge \, {{\mathbb {P}}{}}({\text {Bin}}(m_1,1-4\xi ) \ge m_0) \, \ge \, 1- n^{-\omega (r)}, \end{aligned} \end{aligned}

where we used standard Chernoff bounds and that $$m_1 (1-4\xi ) > m_0/(1-\xi )$$ for $$n \ge n_0(\xi )$$. Hence

\begin{aligned} {{\mathbb {P}}{}}(\chi '({\mathcal {H}}_{m_0}^*) \ge c) \, \le \, {{\mathbb {P}}{}}(\chi '({\mathcal {H}}_{m_1}) \ge c) + n^{-\omega (r)} \, \le \, n^{-\omega (r)} , \end{aligned}
(81)

where we invoked Theorem 2 with m set to $$m_1$$ (which applies since $${n^{1+\sigma } \le m} \le m_1< {4 \xi e({\mathcal {H}}) < n^r}$$). This completes the proof by combining (80) and (81) with $$c \le (1+2\delta )rm/n$$. $$\square$$

### C Heuristics: random greedy edge coloring algorithm

In this appendix we give, for the greedy coloring algorithm from Sect. 3, two heuristic explanations for the trajectories $$|Q_{e}(i)| \approx {\hat{q}}(t)$$ and $$|Y_{v,c}(i)| \approx {\hat{y}}(t)$$ that these random variables follow, where $${t = t(i,m) = i /m}$$.

For our first pseudo-random heuristic, we write $$E_i=\{e_1, \ldots , e_i\}$$ for the multi-set of edges appearing during the first i steps of the algorithm. Ignoring that edges can appear multiple times, our pseudo-random ansatz is that the edges in $$E_i$$ and their assigned colors are approximately independent with

\begin{aligned} {{\mathbb {P}}{}}\bigl (\text {e in} E_i\, \text {and colored}\,c\bigr ) \approx \frac{|E_i|}{|E({\mathcal {H}})|} \cdot \frac{1}{q} \approx \frac{i}{nD/r} \cdot \frac{1}{rm/n} = \frac{t}{D} =: p(t,D) = p, \end{aligned}

where independence only holds with respect to colorings that are proper, i.e., possible in the algorithm. Using this heuristic ansatz, we now consider the event $${\mathcal {E}}_{v,c}$$ that no edge $$f \in E_i$$ with $$v \in f$$ is colored c. Exploiting that no two distinct edges containing v can receive the same color in the algorithm (since this coloring would not be proper), our pseudo-random ansatz and the degree assumption (2) then suggests that

\begin{aligned} {{\mathbb {P}}{}}(\lnot {\mathcal {E}}_{v,c}) = \sum _{f \in E({\mathcal {H}}): v \in f} \hspace{-0.25em} {{\mathbb {P}}{}}(f \hbox {in}\, E_i\, \text {and colored}~c) \approx D \cdot p = t. \end{aligned}

Since for every pair of vertices there are only at most $$n^{-\sigma }D$$ edges containing both (by the codegree assumption), for $$\ell = o(\log n)$$ distinct vertices $$v_{1},\ldots ,v_{\ell }$$ our pseudo-random ansatz also loosely suggests that

\begin{aligned} {{\mathbb {P}}{}}\Bigl ( \bigcap _{i \in [\ell ]}{\mathcal {E}}_{v_i,c}\Bigr ) \approx \prod _{i \in [\ell ]}{{\mathbb {P}}{}}({\mathcal {E}}_{v_i,c}) + O\bigl (\ell ^2 \cdot n^{-\sigma }D \cdot p\bigr )\approx (1-t)^\ell . \end{aligned}

Recalling (46) from Sect. 3, using linearity of expectation we then anticipate $$|Q_{e}(i)| \approx {\hat{q}}(t)$$ based on

\begin{aligned} {{\mathbb {E}}{}}|Q_{e}(i)| = \sum _{c \in [q]}{{\mathbb {P}}{}}\bigl (c \in Q_{e}(i)\bigr ) = \sum _{c \in [q]}{{\mathbb {P}}{}}\Bigl ( \bigcap _{v \in e}{\mathcal {E}}_{v,c}\Bigr ) \approx q \cdot (1-t)^r= {\hat{q}}(t). \end{aligned}

Mimicking this reasoning, recalling (47) we similarly anticipate $$|Y_{v,c}(i)| \approx {\hat{y}}(t)$$ based on

\begin{aligned} {{\mathbb {E}}{}}|Y_{v,c}(i)| = \sum _{f \in E({\mathcal {H}}): v \in f} \hspace{-0.25em} {{\mathbb {P}}{}}\bigl (c \in Q_{f \setminus \{v\}}(i)\bigr ) \approx D \cdot (1-t)^{r-1} = {\hat{y}}(t). \end{aligned}

In our second expected one-step changes heuristic we assume for simplicity that there are deterministic approximations $$|Q_{e}(i)| \approx f(t) q$$ and $$|Y_{v,c}(i)| \approx g(t) D$$. Using these approximations and $$q \approx rm/n$$, the calculations leading to (55)–(56) and (58)–(59) in Sect. 3.2.1 then suggest that

\begin{aligned} {{\mathbb {E}}{}}\bigl (|Q_{e}(i+1)|-|Q_{e}(i)| \, \big | \, {\mathcal {F}}_i\bigr )&\approx - \frac{f(t) q \cdot r\cdot g(t)D}{nD/r\cdot f(t) q} \approx -\frac{rg(t)q}{m} , \end{aligned}
(82)
\begin{aligned} {{\mathbb {E}}{}}\bigl (|Y_{v,c}(i+1)|-|Y_{v,c}(i)| \, \big | \, {\mathcal {F}}_i\bigr )&\approx - \frac{g(t)D \cdot (r-1) \cdot g(t)D}{nD/r\cdot f(t) q} \approx -\frac{(r-1)g^2(t)D}{f(t) m} , \end{aligned}
(83)

where $${\mathcal {F}}_i$$ denotes the natural filtration of the algorithm after i steps. Since the left-hand sides of (82)–(83) are approximately equal to $$[f(t+1/m)-f(t)]q \approx f'(t)q/m$$ and $$g'(t)D/m$$, respectively, we anticipate

\begin{aligned} f'(t) = -rg(t) \quad \text { and } \quad g'(t) = -(r-1) g^2(t)/f(t). \end{aligned}
(84)

Noting $$|Q_{e}(0)| = q$$ and $$|Y_{v,c}(0)| \approx D$$, we also anticipate $$f(0)=g(0)=1$$. The solutions $${f(t)=(1-t)^r}$$ and $${g(t)=(1-t)^{r-1}}$$ then make $$|Q_{e}(i)| \approx f(t) q = {\hat{q}}(t)$$ and $$|Y_{v,c}(i)| \approx g(t) D = {\hat{y}}(t)$$ plausible.

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Guo, H., Patton, K. & Warnke, L. Prague Dimension of Random Graphs. Combinatorica 43, 853–884 (2023). https://doi.org/10.1007/s00493-023-00016-9

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