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Hamiltonicity of Step-graphons

Xudong Chen

TL;DR

This work analyzes Hamiltonicity in directed random graphs generated from step-graphons, proving a zero-one law for the probability of containing a Hamiltonian cycle or decomposition as the graph order grows. It introduces the concentration vector $x^*$, skeleton graph $ ec{S}$, node-cycle incidence $Z$, and node-flow cone $ obreak extsf{X}$ as the core determinants of the limiting behavior, and formulates four partition-invariant conditions (A,B,B',C) that decisively predict the limit. The authors provide illustrative numerical validation, a detailed proof strategy combining S-partite graph theory, edge-flow cones, and the Blow-up Lemma, and a constructive framework to embed Hamiltonian structures into large random digraphs. The results extend previous symmetric-case analyses to general, asymmetric step-graphons and have implications for network controllability and topology-synthesis under uncertainty.

Abstract

In this paper, we sample directed random graphs from (asymmetric) step-graphons and investigate the probability that the random graph has at least a Hamiltonian cycle (or a node-wise Hamiltonian decomposition). We show that for almost all step-graphons, the probability converges to either zero or one as the order of the random graph goes to infinity--we term it the zero-one law. We identify the key objects of the step-graphon that matter for the zero-one law, and establish a set of conditions that can decide whether the limiting value of the probability is zero or one.

Hamiltonicity of Step-graphons

TL;DR

This work analyzes Hamiltonicity in directed random graphs generated from step-graphons, proving a zero-one law for the probability of containing a Hamiltonian cycle or decomposition as the graph order grows. It introduces the concentration vector , skeleton graph , node-cycle incidence , and node-flow cone as the core determinants of the limiting behavior, and formulates four partition-invariant conditions (A,B,B',C) that decisively predict the limit. The authors provide illustrative numerical validation, a detailed proof strategy combining S-partite graph theory, edge-flow cones, and the Blow-up Lemma, and a constructive framework to embed Hamiltonian structures into large random digraphs. The results extend previous symmetric-case analyses to general, asymmetric step-graphons and have implications for network controllability and topology-synthesis under uncertainty.

Abstract

In this paper, we sample directed random graphs from (asymmetric) step-graphons and investigate the probability that the random graph has at least a Hamiltonian cycle (or a node-wise Hamiltonian decomposition). We show that for almost all step-graphons, the probability converges to either zero or one as the order of the random graph goes to infinity--we term it the zero-one law. We identify the key objects of the step-graphon that matter for the zero-one law, and establish a set of conditions that can decide whether the limiting value of the probability is zero or one.

Paper Structure

This paper contains 27 sections, 26 theorems, 158 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main Result
  3. Key objects
  4. Necessary and sufficient conditions for (strong) $H$-property
  5. Illustration and numerical validation
  6. Sketch of proof for the Main Theorem
  7. On necessity of Conditions $A$, $B'$, and $C$
  8. On sufficiency of Conditions $A$, $B$, and $C$
  9. On Necessity of Conditions $A$, $B'$, and $C$
  10. Pre-processing: Removal of self-loops
  11. Proof for Condition $C$
  12. Proof for Condition $A$
  13. Proof for Condition $B$
  14. Hamiltonicity of complete $\vec{S}$-partite graphs
  15. Proof of item 1
  16. ...and 12 more sections

Key Result

Lemma 1

Let $B_{\vec{S}_p}$, for $p = 1,\ldots, q$ be the bipartite graph associated with $\vec{S}_p$. Let $\tau_p$ be the number of connected components of $B_{\vec{S}_p}$. Then, In particular, $\operatorname{co-rank}(Z) = 0$ if and only if every $B_{\vec{S}_p}$ is connected.

Figures (9)

  • Figure 1: The step-graphon $W$ in (a) has the partition sequence $\sigma=\frac{1}{16}(0,1,4,9,16)$. The value of $W$ is shade coded, with black being $1$ and white being $0$. The digraph $\vec{S}$ in (b) is the skeleton graph associated with $W$ with respect to the partition $\sigma$. The digraph $\vec{G}_{n}$, with $n = 10$, in (c) is sampled from $W$. It has a Hamiltonian decomposition, highlighted in blue, which comprises a $4$-cycle and three $2$-cycles.
  • Figure 2: The bipartite graph $B_{\vec{S}}$ in (b) is associated with the digraph $\vec{S}$ in (a). Undirected edges $(u'_i,u"_j)\in E(B_{\vec{S}})$ one-to-one correspond to the directed edges $u_iu_j\in E(\vec{S})$.
  • Figure 3: Left: Four step-graphons and the associated skeleton graphs, where $\vec{S}$ in (e) corresponds to $W_a$, $W_b$, $W_c$, and $\vec{S}'$ in (f) corresponds to $W_d$. Right: The empirical probability $p(n)$ that $\vec{G}_n\sim W_\star$ has a Hamiltonian decomposition, with $20,000$ samples for each $n = 10,50,100,500,1000,2000,5000$.
  • Figure 4: The step-graphon $W'$ in (b) is obtained from $W$ in (a) by first subdividing the right-bottom block into $2$-by-$2$ sub-blocks and then setting the value of the two diagonal sub-blocks to zero. A partition sequence $\sigma$ for $W$ is $\sigma = \frac{1}{16}(0,1,4,9,16)$. The subdivision then gives rise to partition sequence $\sigma' = \frac{1}{16}(0,1,4,9,12.5,16)$ for $W'$. The two digraphs $\vec{S}$ and $\vec{S}'$ shown in (c) and (d) are the skeleton graphs associated with $W$ and $W'$, respectively. The digraph $\vec{S}'$ can be obtained from $\vec{S}$ by removing the self-loop $u_4u_4$ and by adding the node $u_5$ and the edges $u_5 u_1$, $u_5u_3$, $u_3u_5$, $u_5u_4$, and $u_4u_5$, which are highlighted in red---we call this procedure a surgery of $\vec{S}$ on node $u_4$.
  • Figure 5: The bipartite graph $B_{\vec{S}}$ in (a) (resp., $B_{\vec{S}'}$ in (b)) is associated with $\vec{S}$ in \ref{["sfig1:ss'"]} (resp., $\vec{S}'$ in \ref{["sfig2:ss'"]}). The graph $B_{\vec{S}'}$ can be obtained from $B_{\vec{S}}$ by removing the edge $(u'_4, u"_4)$ and by adding nodes $u'_5$ and $u"_5$ and edges $(u'_5, u"_1)$, $(u'_5, u"_3)$, $(u'_3, u"_5)$, $(u'_5, u"_4)$, and $(u'_4, u"_5)$, highlighted in red.
  • ...and 4 more figures

Theorems & Definitions (57)

  • Definition 1: Step-graphon
  • Definition 2: $H$-property
  • Definition 3: Concentration vector
  • Definition 4: Skeleton graph
  • Definition 5: Node-cycle incidence vector/matrix
  • Lemma 1
  • Definition 6: Node-flow cone
  • Proposition 1
  • Definition 7
  • Definition 8: $\vec{S}$-partite graph
  • ...and 47 more