Minimum Vertex-Degree Conditions

• Minimum vertex-degree conditions are threshold values ensuring that every vertex in a graph or hypergraph meets a minimum connection level to force desired structures such as Hamilton cycles and spanning subgraphs.
• These thresholds are instrumental in analyzing extremal properties and are proven using techniques like the Regularity Lemma, absorbing methods, and fractional-to-integral tiling conversions.
• The conditions provide sharp criteria for connectivity, rigidity, and chromatic profiles, influencing both theoretical advancements and practical algorithms in graph optimization.

Minimum vertex-degree conditions describe the threshold values of the least degree among the vertices in a graph or hypergraph that guarantee the presence of specified substructures (such as spanning subgraphs, Hamilton cycles, factors, or connectivity properties). These conditions serve as sharp combinatorial and extremal thresholds across diverse settings, including simple graphs, bipartite graphs, and uniform hypergraphs, and underpin much of modern extremal combinatorics.

1. Classical Thresholds and Spanning Structures

The archetype is Dirac’s theorem: every $n$-vertex graph $G$ with $\delta(G)\ge n/2$ is Hamiltonian. This principle generalizes across a spectrum of properties, with explicit focus on minimum vertex-degree thresholds for spanning regular subgraphs, perfect tilings, and connectivity.

A central recent result is the optimal minimum degree for forcing a spanning $r$-regular, $r$-connected subgraph: there exists $n_0$ such that every $n$-vertex $G$ with

$$\delta(G) \geq \frac{n+r-2}{2}, \qquad n\ge n_0, \; nr\equiv0\pmod2$$

contains a spanning $r$-regular, $r$-connected subgraph, with the bound being sharp (Hahn-Klimroth et al., 2021). The proof employs a trichotomy of cases: (A) non-extremal ("no sparse cut")—using Szemerédi’s Regularity Lemma with the Blow-Up Lemma to globally embed the structure; (B) a near-bipartite extremal case; and (C) an almost-clique extremal case, each reinforced by absorbing arguments and explicit constructions.

In the bipartite context, the minimum vertex-degree sum condition for $K_{s,s}$-tiling exhibits three distinct regimes, depending on asymmetry $D=|\delta_U-\delta_V|$ (Czygrinow et al., 2013):

• For small $D$, the threshold is $\delta_U+\delta_V\ge n+3s-5$ and is tight.
• For intermediate $D$, the threshold decreases to $n+2s-2\sqrt{s}+c(s)$, still best possible.
• If $D$ is extreme, perfect tiling can be blocked despite $\delta_U+\delta_V \gg n$.

2. Hamiltonicity and Cycle Structures in Graphs and Hypergraphs

Minimum vertex-degree constraints define critical thresholds for Hamiltonicity and more intricate spanning cycles.

For $3$-uniform hypergraphs, the exact threshold for loose Hamilton cycles is

$$\delta_1(H)\ge \binom{n-1}{2} - \binom{\lfloor3n/4\rfloor}{2} + c\,, \;\; c = \begin{cases} 2 & n \equiv 0\pmod{4} \ 1 & n \equiv 2\pmod{4} \end{cases}$$

and is tight (Han et al., 2013). The methodology fuses the absorbing method with regularity and path-tiling (notably the $Y$-tiling lemma), precisely partitioning the extremal and non-extremal cases.

For tight Hamilton cycles, the minimum vertex-degree threshold is asymptotically $(5/9)\binom{n}{2}$, and for loose Hamilton cycles, $(7/16)\binom{n}{2}$, both tight up to $o(n^2)$ corrections (Reiher et al., 2016, Buß et al., 2016). The proofs in each employ robust linking via absorbing paths, matchings in link graphs, and fractional–to–integral tiling conversions.

The Dirac and Ore thresholds extend to local versions: A graph is locally Dirac (resp., locally Ore) if each open neighborhood induces a subgraph satisfying the Dirac (resp., Ore) condition. These local conditions yield global properties, such as $3$-vertex-connectivity, equality of edge- and vertex-connectivity (for locally Dirac), and pancyclic or cycle-extendable behavior under maximal degree constraints (Kubicka et al., 2015).

3. Degree Conditions for Tiling, Packing, and Extremal Substructures

For tiling complete multipartite graphs in $3$-uniform hypergraphs, the minimum vertex-degree threshold for a perfect $K_{a,b,c}$-tiling is

$$t_1(n,K_{a,b,c}) = (\alpha(a,b,c)+o(1))\binom{n}{2}$$

where

$$\alpha(a,b,c) = \max\left\{ f(a,b,c),\, 1-(\tfrac{b+c}{k})^2,\, (\tfrac{a+b}{k})^2 \right\}$$

with $k=a+b+c$ and $f(a,b,c)$ given by explicit lattice and covering barriers (Han et al., 2015). A lattice-based absorbing method, together with a fractional homomorphism-tiling → integral tiling conversion and shadow lemma, yields the asymptotic threshold.

Vertex-degree sum conditions, as in $\sigma_2(G)$ (minimum degree sum over nonadjacent pairs), sharpen or extend Dirac-type theorems by interpolating between local and global hypotheses. For example, if $\sigma_2(G)\ge 2(s_1+s_2+1)-1$, $G$ contains two disjoint subgraphs each with high $\sigma_2$ (Chiba et al., 2015).

4. Rigidity and Structural Graph Properties

Minimum vertex-degree (and degree sum) conditions exhibit critical thresholds for rigidity in Euclidean spaces. For $d$-dimensional generic rigidity,

$$f(n,d) = \min\{k: \delta(G)\ge k \implies G \text{ is rigid in } \mathbb{R}^d\}$$

satisfies $f(n,d) = \lceil (n+d-2)/2 \rceil$ for $n\ge29d$, and this is sharp (Jordán et al., 29 Oct 2025). In terms of degree-sum,

$$g(n,d) = n+d-2 \quad \text{for} \quad n\ge d(d+2)\,.$$

For small $d$ one obtains explicit cases: $f(n,2) = \lceil n/2 \rceil$, $f(n,3)=\lceil (n+1)/2 \rceil$, with classified exceptions for small graphs. The approach is matroidal, leveraging coning, rank parameters, and rank-contribution functions.

Open problems involve sharpening the approximate bounds for $d=O(n/\log^2 n)$, as in (Krivelevich et al., 2024), and closing the factor-2 gap for large $d$.

5. Chromatic Profiles, Rainbow Structures, and Local Conditions

The interplay between forbidden substructures and minimum vertex-degree gives rise to "chromatic profile" functions, e.g., $\delta_\chi(\mathcal{F},c)$: the infimum $\alpha$ such that every $\mathcal{F}$-free graph with $\delta(G)\ge\alpha n$ has $\chi(G)\le c$. For odd cycles $C_{2k+1}$, it is determined that

$$\delta_\chi(C_{2k+1},c) = \frac{1}{2c+2}, \quad k\ge 3c+4$$

with extremal examples given by balanced blow-ups of $K_{c+1}$ (Yan et al., 2024). The central tool is the "strong $2k$-core"—a core subgraph with both odd- and even-diameter at most $2k$.

Rainbow structures in edge-colored settings exhibit discrete threshold phenomena. For three graphs $(G_1, G_2, G_3)$ on a common $n$-vertex set, a "threshold-triple" minimum degree condition

$$\delta_1 > (1-1/r)n,\quad \delta_2+\delta_3 \ge 2n/r$$

guarantees the existence of a rainbow triangle, with sharp Turán-type extremal constructions showing tightness (Falgas-Ravry et al., 2023).

6. Advanced Topics: Proper-Path Connectivity, Knitted Graphs, and Local–Global Convergence

In edge-colored graphs, a blend of minimum degree and edge count yields precise thresholds for the proper connection number: the minimal color count needed for proper-path connectivity. The function

$$|E(G)| \ge \binom{n-m-(k+1-m)(\delta+1)}{2} + (k+1-m)\binom{\delta+1}{2} + k+2$$

(with $m$ determined by $k$ and $\delta$) ensures $pc(G)\le k$, with explicit constructions showing sharpness (Guan et al., 2018).

Knitted graphs generalize $k$-ordered and linked graphs; Liu–Rolek–Yu established that $\delta(G)\ge (n+k)/2-1$ (for $n\ge2k+3$) suffices for $G$ to be $k$-knitted, and this bound is sharp (Liu et al., 2018). Advanced connectivity results follow, such as every $k$-contraction-critical graph being at least $\lceil k/8\rceil$-connected.

For vertex-connectivity properties, a suite of results yields sufficient conditions for $k$-connectedness, maximal connectivity, and super-connectivity in terms of $\delta(G)$, edge count $m$, and spectral radius $\rho(G)$, with exact (and uniquely extremal) threshold graphs classified (Hong et al., 2017).

Table: Sharp Minimum Vertex-Degree Thresholds for Representative Problems

Structure/Property	Minimum vertex-degree threshold	Comments / Reference
Hamilton cycle (graphs)	$\delta(G)\ge n/2$	Dirac’s theorem
$r$-regular, $r$-connected subgraph	$\delta(G)\ge (n+r-2)/2$	Tight (Hahn-Klimroth et al., 2021)
$K_{s,s}$-tiling (balanced bipartite)	small $D$: $\delta_U+\delta_V\ge n+3s-5$	Varies with $D$ (Czygrinow et al., 2013)
Loose Hamilton cycle ($3$-graph)	$$\delta_1(H)\ge \binom{n-1}{2}-\binom{\lfloor 3n/4\rfloor}{2}+c$$	Tight (Han et al., 2013)
Tight Hamilton cycle ($3$-graph)	$\delta_1(H)\ge (5/9+o(1))\binom{n}{2}$	Asymptotic (Reiher et al., 2016)
Perfect $K_{a,b,c}$-tiling ($3$-graph)	$\delta_1(H)\ge (\alpha(a,b,c)+o(1))\binom{n}{2}$	Asymptotic (Han et al., 2015)
$k$-Knitted ($n\ge 2k+3$)	$\delta(G)\ge (n+k)/2-1$	Tight (Liu et al., 2018)
$d$-rigid ($n\ge 29d$)	$\delta(G)\ge \lceil (n+d-2)/2\rceil$	Tight (Jordán et al., 29 Oct 2025)
Proper connection number $\leq k$	See size/min-degree formula above	(Guan et al., 2018)

These sharp thresholds and exact constructions underpin a unified theory of extremal and structural graph properties, with ongoing research extending to hypergraphs of higher uniformity, locally enforced conditions, and random and colored settings. Each result not only exposes the fine structure that minimum degree controls but provides the blueprint for stability and robustness phenomena in combinatorial optimization and graph algorithms.