June 01, 2026
For a given subset \(S\subseteq V(G)\) of a graph \(G\), the pair \((G,S)\) is knitted if for every partition of \(S\) into non-empty subsets \(S_1, S_2, \ldots, S_t\), there exist pairwise disjoint connected subgraphs \(C_1, C_2, \ldots, C_t\) in \(G\) such that \(S_i\subseteq V(C_i)\) for all \(1 \le i \le t\). A graph \(G\) is \(\ell\)-knitted if \((G,S)\) is knitted for every subset \(S\subseteq V(G)\) of size \(\ell\). In this paper, we prove that every \(8\ell\)-connected graph is \(\ell\)-knitted. We subsequently apply this result to Hadwiger’s Conjecture, which states that every \(k\)-chromatic graph contains a \(K_k\)-minor. Specifically, we demonstrate that the vertex connectivity of any minimal counterexample to Hadwiger’s Conjecture is at least \(\lceil k/8 \rceil\), improving upon the previous lower bound of \(\lceil 2k/27 \rceil\) established by Kawarabayashi (2007). Our proof corrects a gap in the argument of Kawarabayashi-Yu (2013) and establishes the claim stated without proof in Liu–Rolek–Yu (2019).
One of the most profound problems in graph theory is Hadwiger’s Conjecture, which posits that every \(k\)-chromatic graph contains a \(K_k\)-minor. A graph \(H\) is a minor of a graph \(G\) if \(H\) can be obtained from a subgraph of \(G\) by contracting edges.
It is well known that Hadwiger’s Conjecture holds for \(k\le 6\). In 1937, Wagner [1] proved that the case \(k=5\) is equivalent to the Four Color Theorem. Nearly 60 years later, Robertson, Seymour, and Thomas [2] proved that the case \(k=6\) is also equivalent to the Four Color Theorem. In their proof, minimal counterexamples—often referred to as “contraction-critical non-complete graphs”—play a foundational role.
Kawarabayashi and Toft [3] showed that any \(7\)-chromatic graph contains either a \(K_7\)-minor or a \(K_{4,4}\)-minor, a result for which the connectivity properties of minimal counterexamples are once again pivotal. Extensive research has been dedicated to the connectivity properties of contraction-critical graphs. Dirac [4] proved that every \(k\)-contraction-critical graph is \(5\)-connected for \(k\ge 5\). Mader [5] extended this deep structural result, establishing that every \(k\)-contraction-critical graph is \(7\)-connected for \(k\ge 7\), and every \(6\)-contraction-critical graph is \(6\)-connected. Toft [6] proved that \(k\)-contraction-critical graphs are \(k\)-edge-connected. Kawarabayashi [7] established the first general linear bound on the vertex connectivity of minimal counterexamples to Hadwiger’s Conjecture:
Theorem 1 (Kawarabayashi [7]). For all positive integers \(k\), every minimal (with respect to the minor relation) \(k\)-chromatic counterexample to Hadwiger’s Conjecture is \(\lceil \frac{2k}{27}\rceil\)-connected.
The primary tool used in the proof of Theorem 1 was the concept of \(k\)-linked graphs. A graph \(G\) is \(k\)-linked if for every \(2k\) distinct vertices \(u_1, v_1, u_2, v_2, \ldots, u_k,v_k\), there exist \(k\) disjoint paths \(P_1, P_2, \ldots, P_k\) such that \(P_i\) connects \(u_i\) and \(v_i\). The study of \(k\)-linked graphs has a rich history and remains central to structural graph theory.
In this paper, we improve the bound in Theorem 1 by investigating a related notion called a “knitted graph,” introduced by Bollobás and Thomason [8]. For \(1\le m\le k\le |V(G)|\), a graph is \((k,m)\)-knit if, whenever \(S\) is a set of \(k\) vertices of \(G\) and \(S_1, \ldots, S_t\) is a partition of \(S\) into \(t\ge m\) non-empty parts, \(G\) contains vertex-disjoint connected subgraphs \(C_1, \ldots, C_t\) such that \(S_i\subseteq V(C_i)\) for \(1\le i\le t\). Clearly, a \((2k, k)\)-knit graph is \(k\)-linked. In [8], Bollobás and Thomason proved that if a \(k\)-connected graph \(G\) contains a minor \(H\) with minimum degree at least \(0.5(|V(H)|+\lfloor 5k/2\rfloor-2-m)\), then \(G\) is \((k, m)\)-knit. They used this to show that \(22k\)-connected graphs are \(k\)-linked, providing the first linear upper bound on the connectivity required for a graph to be \(k\)-linked.
We consider a natural generalization of the \((k,m)\)-knit property.
Definition 1. For a subset \(S\subseteq V(G)\), the pair \((G, S)\) is knitted* if for every partition of \(S\) into non-empty subsets \(S_1, S_2, \ldots, S_t\), there exist pairwise disjoint connected subgraphs \(C_1, C_2, \ldots, C_t\) in \(G\) such that \(S_i\subseteq V(C_i)\). A graph \(G\) is \(\ell\)-knitted if \((G,S)\) is knitted for all \(S\subseteq V(G)\) with \(|S|=\ell\).*
It is clear that an \(\ell\)-knitted graph is \((\ell, m)\)-knit for all \(m\le \ell\). In this paper, we establish a connectivity threshold for a graph to be \(\ell\)-knitted.
Definition 2. A pair \((A, B)\) of vertex sets is a separation* of \(G\) if \(V(G)=A\cup B\) and there are no edges between \(A\setminus B\) and \(B\setminus A\). The order of a separation \((A, B)\) is \(|A\cap B|\). If \(S\subseteq A\), we say that \((A,B)\) is a separation of \((G,S)\).*
Our main structural result is the following:
Theorem 2. Let \(k\) and \(\ell\) be positive integers, and let \(S\subseteq V(G)\) with \(|S|=\ell<k/8\). If there is no separation of \((G,S)\) of order less than \(\ell\), and every vertex in \(G-S\) has degree at least \(k-1\), then \((G,S)\) is knitted.
Note that in [9], we claimed a proof of the above result for \(|S|=\ell<k/9\). However, we later discovered a gap in the argument, arising from an overly strong conclusion asserted in Theorem 6. Subsequently, by replacing the sufficient degree condition used in [9] with the stronger condition established in [10], we claimed, without proof, that the above result also holds for \(|S|=\ell<k/8\) in [10].
Theorem 9 on edge-density (presented in Section 3) is logically stronger than Theorem 2 and will serve as our primary tool. We are now ready to state and prove our main result on the connectivity of minimal counterexamples to Hadwiger’s Conjecture.
Theorem 3. For all positive integers \(k\), every \(k\)-chromatic minimal (with respect to the minor relation) counterexample to Hadwiger’s Conjecture is \(\lceil \frac{k}{8}\rceil\)-connected.
Proof. Assume, for the sake of contradiction, that the statement fails. Let \(G\) be a minimal \(k\)-chromatic graph that contains no \(K_k\)-minor and has vertex connectivity strictly less than \(k/8\). Let \(S\) be a minimum vertex cutset of \(G\). Thus, \(|S|=\ell < k/8\). Let \(A_1\) be the vertex set of a connected component of \(G-S\), and let \(A_2 = V(G) \setminus (S \cup A_1)\).
By the minimality of \(G\), both induced subgraphs \(G[A_1\cup S]\) and \(G[A_2\cup S]\) are proper subgraphs, and therefore have chromatic number at most \(k-1\).
We partition \(S\) into independent sets greedily: let \(S_1\) be a maximal independent set in \(G[S]\), and for \(i\ge 2\), let \(S_i\) be a maximal independent set in the induced subgraph \(G[S\setminus \bigcup_{j=1}^{i-1} S_j]\). Let \(S_1, S_2, \ldots, S_t\) be the resulting partition.
Observe the structural implication of this greedy maximal choice: for any \(i > j\), every vertex in \(S_i\) must have at least one neighbor in \(S_j\), because adding any vertex of \(S_i\) to \(S_j\) would violate the independence of \(S_j\). Consequently, if we find pairwise disjoint connected subgraphs \(C_1, \ldots, C_t\) in \(G\) such that \(S_i \subseteq V(C_i)\), contracting each \(C_i\) into a single vertex will form a clique \(K_t\) on these contracted vertices.
Since \(G\) is a minimal \(k\)-chromatic graph, its minimum degree is at least \(k-1\). Thus, every vertex in \(A_p\) (for \(p \in \{1,2\}\)) has at least \(k-1\) neighbors in \(A_p\cup S\). Note also that any separation of \((A_1\cup S, S)\) or \((A_2\cup S, S)\) is a valid separation of \((G,S)\). Since \(S\) is a minimum cutset of \(G\), neither \((A_1\cup S, S)\) nor \((A_2\cup S,S)\) can have a separation of order less than \(\ell\).
Therefore, the pairs \((A_1\cup S, S)\) and \((A_2\cup S, S)\) both satisfy the conditions of Theorem 2. By Theorem 2, both pairs are knitted.
This guarantees the existence of disjoint connected subgraphs \(C_i \subseteq A_1\cup S\) such that \(S_i\subseteq V(C_i)\), and similarly, disjoint connected subgraphs \(D_i \subseteq A_2\cup S\) such that \(S_i\subseteq V(D_i)\). We contract the components \(C_i\) within \(A_1\cup S\) into single vertices; by our earlier observation, these contracted vertices form a complete graph on the sets \(S_1, S_2, \dots, S_t\). Let \(G_1\) be the graph obtained from this contraction combined with the uncontracted vertices of \(A_2\). Similarly, we can contract the components \(D_i\) within \(A_2\cup S\) to form a complete graph on the same partition sets, yielding a graph \(G_2\) combined with \(A_1\).
Both \(G_1\) and \(G_2\) are proper minors of \(G\). By the minimality of \(G\), we have \(\chi(G_1) \le k-1\) and \(\chi(G_2) \le k-1\). Crucially, the contracted vertices corresponding to \(S_1, \dots, S_t\) form a clique in both \(G_1\) and \(G_2\). Therefore, any \((k-1)\)-coloring of \(G_1\) and \(G_2\) must assign all \(t\) contracted vertices strictly distinct colors. Since the cliques in \(G_1\) and \(G_2\) represent the exact same partition of \(S\), we can naturally permute the colors of \(G_2\) so that they perfectly match the colors assigned to the clique in \(G_1\). Fusing these colorings yields a valid proper \((k-1)\)-coloring for the entire graph \(G\), contradicting the assumption that \(G\) is \(k\)-chromatic. This completes the proof. ◻
The remainder of the paper is dedicated to proving Theorem 2. We achieve this in two steps: first (Section 3), we establish that the graphs under study are either knitted or contain a highly dense subgraph; second (Section 4), we demonstrate that such dense subgraphs inherently contain a knitted subgraph. This approach structurally mirrors the techniques utilized by Thomas and Wollan [11].
In this section, we establish the following results concerning the interaction between a forest with a prescribed boundary set and a vertex outside the forest. Although the result is developed for use in our proof, it may also be of independent interest.
Theorem 4 (General Reduction). Let \(W\) be a graph, and let \(S' \subseteq V(W)\) be a boundary set with \(|S'| \ge 2\). Let \(F\) be a forest in \(W\) with \(c \ge 1\) connected components such that \(S' \subseteq V(F)\) and all leaves of \(F\) belong to \(S'\). Let \(u \in V(W) \setminus V(F)\). Suppose that \(d(u, F) \ge |S'| + k\), where \(k = 2\) if \(c = 1\), and \(k = 1\) if \(c \ge 2\). Then the induced subgraph \(W[V(F) \cup \{u\}]\) contains a forest \(F_0\) with \(u \in V(F_0)\) such that \(S' \subseteq V(F_0)\), all leaves of \(F_0\) in \(S'\), and at least one of the following holds:
\(|V(F_0)| < |V(F)|\)
\(|V(F_0)| = |V(F)|\) and \(c(F_0) < c(F)\)
Proof. Partition \(S'\) across the components \(T_1 \dots T_c\) of \(F\) as \(S'_i = S' \cap V(T_i)\). Since all leaves of \(F\) are in \(S'\), each \(T_i\) is minimally spanned by \(S'_i\). Let \(N = N_W(u) \cap V(F)\). We have \(|N| \ge |S'| + k\). Let \(I\) be the index set of components containing at least one vertex of \(N\).
For each \(i \in I\) and \(s \in S'_i\), let \(P_s\) be a shortest path in \(T_i\) from \(s\) to \(N\). Let \(y_s \in N\) be its terminus. Define \(F_{\text{ess}}^{(i)} = \bigcup_{s \in S'_i} P_s\), and let \(Y_i = \{y_s \mid s \in S'_i\}\). Let \(Y = \bigcup Y_i\). Clearly, \(|Y| \le |S'|\).
We construct an intermediate graph \(F'_0\): \[F'_0 = \left( \bigcup_{i \in I} F_{\text{ess}}^{(i)} \right) \cup \{u\} \cup \{uy \mid y \in Y\} \cup \left( \bigcup_{j \notin I} T_j \right)\] While \(F'_0\) connects the active components through \(u\) and spans \(S'\), it might contain cycles if paths in \(F_{\text{ess}}^{(i)}\) coalesce and then diverge to multiple targets in \(Y_i\). We resolve this by taking \(F_0\) to be a minimal spanning sub-forest of \(F'_0\) with respect to \(S'\). By minimality, \(F_0\) is cycle-free and all leaves of \(F_0\) are in \(S'\).
We must guarantee \(u \in V(F_0)\). If \(u\) were removed during minimization, \(F_0\) would be entirely contained within \(F \setminus N\). This implies \(\bigcup F_{\text{ess}}^{(i)}\) inherently spans \(S'\), which forces \(F_{\text{ess}}^{(i)} = T_i\) and \(N = Y\). However, our threshold guarantees \(|N| \ge |S'| + 1\), while \(|Y| \le |S'|\), a contradiction. Thus, \(u\) survives minimization.
Because \(F_0 \subseteq F'_0\), any neighbor of \(u\) in \(N \setminus Y\) is completely excluded from \(F_0\).
If \(c = 1\), \(k=2\). Thus \(|N \setminus Y| \ge |S'|+2 - |S'| = 2\). The order strictly decreases: \(|V(F_0)| \le |V(F)| - 2 + 1 < |V(F)|\), satisfying the requirement of our theorem.
If \(c \ge 2\), \(k=1\). If \(|I|=1\), then \(d(u, T_i)\ge |S'|+1\ge |S_i'|+2\) for some sole component \(T_i\). It follows that \(W[T_i\cup \{u\}]\) has a smaller tree with desired properties. So \(|I| \ge 2\). Then the components merge into one component. Thus \(c(F_0) < c(F)\). Furthermore, \(|V(F_0)| \le |V(F)| - |N \setminus Y| + 1 \le |V(F)|\). ◻
We now classify the exact tight structure when \(F\) is a single tree (\(c=1\)), the degree is lowered to exactly \(d(u, F) = |S'| + 1\), and no reduction is possible. We will not use this result in our proof, but again, it may be of independent interest.
Theorem 5 (Extremal Structure). Let \(F\) be a tree bounding \(S'\) such that \(\text{Leaves}(F) \subseteq S'\). Let \(u \notin V(F)\) such that \(d(u, F) = |S'| + 1\). Suppose \(W[V(F) \cup \{u\}]\) contains NO strictly smaller forest \(F_0\) spanning \(S'\) with \(u \in V(F_0)\) and \(\text{Leaves}(F_0) \subseteq S'\). Then \(F\) is a subdivided star with a unique central vertex \(z\) of degree \(|S'|\), and \(N_W(u) \cap V(F)\) consists exactly of \(z\) and its \(|S'|\) immediate neighbors.
Proof. Let \(N = N_W(u) \cap V(F)\) with \(|N| = |S'| + 1\). For any valid assignment of each \(s \in S'\) to a closest neighbor \(y_s \in N\), we can construct the intermediate graph \(F'_0 = \bigcup P_s \cup \{u\} \cup \{uy_s\}\) and extract its minimal spanning sub-forest \(F_0\).
As established, \(|V(F_0)| \le |V(F)| - |N \setminus Y| + 1\). By the hypothesis that no strictly smaller forest exists, we must have \(|V(F_0)| \ge |V(F)|\) for any valid choice of targets. This forces \(|V(F_0)| = |V(F'_0)| = |V(F)|\), and \(|N \setminus Y| = 1\). Let \(N \setminus Y = \{z\}\). Since \(|N| = |S'|+1\), we must have \(|Y| = |S'|\), meaning each boundary vertex routes to a distinct unique closest neighbor.
Since \(|V(F'_0)| = |V(F)|\), the essential paths must cover exactly \(V(F) \setminus \{z\}\). Removing \(z\) from \(F\) leaves a forest of \(d_F(z)\) components. Because the essential paths cover these components and terminate at exactly \(|S'|\) distinct targets in \(Y\), there are at most \(|S'|\) paths, bounding \(d_F(z) \le |S'|\).
Consider a component \(C\) of \(F \setminus \{z\}\). Because \(F\) is a tree, \(C\) connects to \(z\) via a single unique vertex \(v_C\). For any \(s \in S' \cap V(C)\), the path in \(F\) from \(s\) to \(z\) goes through \(v_C\). Since \(y_s\) is the closest vertex in \(N\) to \(s\), we have \(d(s, y_s) \le d(s, z)\), which expands to \(d(s, v_C) + d(v_C, y_s) \le d(s, v_C) + 1\), forcing \(d(v_C, y_s) \le 1\).
Suppose for contradiction \(y_s \neq v_C\). Then \(d(v_C, y_s) = 1\), making \(y_s\) a branch off \(v_C\). In this layout, \(y_s\) and \(z\) are equidistant from \(s\), causing a tie for the closest neighbor in \(N\). If we alternatively selected \(z\) as the target for \(s\), the new path \(P'_s\) would terminate at \(z\) instead of \(y_s\). Since \(y_s\) is a branch off \(v_C\), the path \(P'_s = s \dots v_C\) strictly excludes \(y_s\). This alternative construction removes \(y_s\) from the essential forest. The resulting new forest \(F^*_0\) would bypass both \(y_s\) and whatever unique vertex represents \(N \setminus Y^*\), yielding \(|V(F^*_0)| \le |V(F)| - 2 + 1 < |V(F)|\). This directly violates our hypothesis that no strictly smaller forest exists.
Therefore, we must have \(y_s = v_C\) universally to prevent tie-breaking reductions. This dictates that \(Y\) is precisely the set of immediate neighbors of \(z\). Since \(Y\) covers all \(|S'|\) distinct paths, \(d_F(z) = |S'|\). Finally, since the components \(C\) are exactly the paths \(P_s\) extending from \(s\) to \(v_C\), \(F\) is structurally identical to a subdivided star centered at \(z\). ◻
We introduce a metric to govern our induction. For a set \(X\subseteq V(G)\), let \(\rho(X)\) denote the number of edges with at least one endpoint in \(X\).
Definition 3. A separation \((A, B)\) of \((G,S)\) is rigid* if the pair \((G[B], A\cap B)\) is knitted.*
Definition 4. Let \(G\) be a graph, \(S\subseteq V(G)\), and \(\alpha>1\) be a real number. The pair \((G, S)\) is \(\alpha \ell\)-massed* if:*
\(\rho(V(G)-S) \ge \lfloor\alpha \ell |V(G)-S|\rfloor\), and
every separation \((A,B)\) of \((G,S)\) of order at most \(|S|-1\) satisfies \(\rho(B\setminus A) \le \alpha \ell |B\setminus A|\).
From now on, we will try to show that \((G,S)\) is \((s_1, \ldots, s_t)\)-knitted, where \(\sum_{i=1}^t s_i=\ell\). Since we do not pose any restriction on the partition, this is equivalent to show that \((G,S)\) is knitted.
Definition 5. The pair \((G,S)\) is \((\alpha, \ell)\)-minimal* if:*
\((G,S)\) is \(\alpha \ell\)-massed,
\(|S|\le \ell\) and \((G,S)\) is not \((s_1, \ldots, s_t)\)-knitted,
subject to the above, \(|V(G)|\) is minimized,
subject to the above, \(\rho(V(G)-S)\) is minimized, and
subject to the above, the number of edges of \(G\) with both endpoints in \(S\) is maximized.
We state the following theorem, which strengthens the core density arguments.
Theorem 6. Let \(\ell\ge 1\) be an integer and \(\alpha\ge 2\) be a real number. If a pair \((G,S)\) is \((\alpha, \ell)\)-minimal, then \(G\) has no rigid separation of order at most \(|S|\). Furthermore, there is a partition of \(S=S_1\cup S_2\ldots\cup S_t\) with \(s_i=|S_i|\), and a subgraph \(H=G[N[v]]\) in \(G\) with \(|V(H)|\le \lceil 2\alpha \ell\rceil\) such that \[d_H(u) \ge \begin{cases} 1+\lfloor\alpha \ell\rfloor-s_i, & if u \in S_i;\\ 1+\lfloor\alpha \ell\rfloor, &if u \in V(H)-S. \end{cases}\]
Proof. Since \((G,S)\) is \((\alpha, \ell)\)-minimal, \((G,S)\) is not \((s_1, \ldots, s_t)\)-knitted. In particular, for a partition \(S_1,\ldots, S_t\) of \(S\) with \(s_i=|S_i|\), we cannot find disjoint connected subgraphs respectively containing \(S_i\)’s. By Definition 5 (5), we may assume that vertices between \(S_i\) and \(S_j\) are all adjacent, since adding edges between \(S_i\) and \(S_j\) maintain \((G,S)\) to be \((\alpha, \ell)\)-minimal.
We structure the proof across three fundamental claims.
Claim 1. \(G\) has no rigid separation of order at most \(|S|\).
Proof. Suppose such a rigid separation \((A,B)\) exists; select one minimizing \(|A|\).
First, assume \(|A\cap B| < |S|\). Let \(G^*[A]\) be the graph obtained by adding all missing edges to make \(A\cap B\) a clique. If \((G^*[A], S)\) is knitted, we can effortlessly transform its knits into a valid knit for \((G,S)\) by leveraging the rigidity of \((A,B)\). Since \((G,S)\) is not knitted, \((G^*[A], S)\) is not knitted. Because \(G\) is \(\alpha \ell\)-massed, \(\rho(B\setminus A) \le \alpha \ell|B\setminus A|\). Consequently, \(\rho(A\setminus S) > \alpha \ell|A\setminus S|\). This means condition (i) of being massed is satisfied, forcing condition (ii) to fail in \(G^*[A]\). Let \((A', B')\) be a separation of \(G^*[A]\) containing \(S \subseteq A'\) that violates (ii), minimizing \(B'\). If \(A\cap B \subseteq A'\), then \((A'\cup B, B')\) acts as a separation in \(G\) violating (ii), which is impossible. Thus, since \(A\cap B\) is a clique, \(A\cap B \subseteq B'\). By the minimality of \(B'\), the subgraph \((G^*[B'], A'\cap B')\) is itself massed and thus knitted. Consequently, \((G^*[B\cup B'], A'\cap B')\) is knitted, making \(A'\cap B'\) a rigid separation of \((G,S)\) strictly smaller than \(A\), contradicting our minimal choice of \(A\).
Next, assume \(|A\cap B| = |S|\). If there are \(|S|\) disjoint paths from \(S\) to \(A\cap B\), the rigidity of \((A,B)\) instantly makes \((G,S)\) knitted. Otherwise, there is a separation \((A'', B'')\) of \((G[A], S)\) of order less than \(|S|\) separating \(S\) from \(A\cap B\). Choosing the minimal \(|A''\cap B''|\), we generate disjoint paths from \(A''\cap B''\) to \(A\cap B\). The rigidity flows back, establishing \((A'', B\cup B'')\) as a rigid separation of \((G,S)\) with \(|A''| < |A|\), again contradicting minimality. ◻
Claim 2. Let \(uv\) be an edge with \(v \notin S\). Then \(d(v) \ge 1+\lfloor\alpha \ell\rfloor\). Moreover, if \(u \notin S\), then \(u\) and \(v\) have at least \(\lfloor\alpha \ell\rfloor\) common neighbors. If \(u \in S\), then \(u\) and \(v\) have at least \(\lfloor\alpha \ell\rfloor - |N(v)\cap S-N[u]|\) common neighbors.
Proof. Let \(G'=G/uv\) be the graph resulting from contracting the edge \(uv\). If \((G', S)\) is knitted, \((G, S)\) inherits the knit. Thus \((G', S)\) must violate (i) or (ii).
Suppose \((G', S)\) violates (i). Then \(\rho(V(G')-S) \le \lfloor\alpha \ell |V(G')-S|\rfloor - 1\). Since \(G\) satisfies (i), \(\rho(V(G)-S)\ge \lfloor\alpha \ell |V(G)-S|\rfloor\). The contraction reduces the vertex count by exactly one. \[\rho(V(G)-S) - \rho(V(G')-S)\ge 1+\lfloor\alpha\ell |V(G)-S|\rfloor - \lfloor\alpha\ell|V(G')-S|\rfloor \ge 1+\lfloor\alpha\ell\rfloor.\] The edges lost in \(\rho(V(G)-S)\) during contraction are exactly the edge \(uv\), one duplicate edge for every common neighbor of \(u\) and \(v\) not in \(S\), and (if \(u \in S\)) any edge from \(v\) to \(S\) that becomes redundant. All such edges are incident to \(v\), establishing \(d(v) \ge \rho(V(G)-S) - \rho(V(G')-S)\ge 1+ \lfloor\alpha\ell\rfloor\). If \(u \notin S\), the exact number of lost edges is \(1 + |N(u) \cap N(v)|\), yielding \(|N(u) \cap N(v)|\ge\lfloor \alpha \ell\rfloor\). If \(u \in S\), the lost edges number \(1 + |N(u) \cap N(v)| - |N(v) \cap S - N[u]|\), preserving the inequality.
Alternatively, assume \((G',S)\) violates (ii). Let \((A',B')\) be the violating separation with minimal \(B'\). By minimality, \((G'[B'], A'\cap B')\) is knitted, meaning \((A', B')\) acts as a rigid separation in \(G'\). In the pre-contraction graph \(G\), this induces a separation \((A,B)\). If \(\{u,v\} \not\subseteq A\cap B\), then \((A,B)\) operates as a rigid separation of \((G,S)\) of order at most \(|S|-1\), decisively violating Claim 1. If \(u,v \in A\cap B\), the minimality of \(B'\) ensures \((G[B], A\cap B)\) is \(\alpha\ell\)-massed and therefore knitted, creating a rigid separation of size \(\le |S|\), violating Claim 1 once more. ◻
Claim 3. Let \(\delta'\) be the minimum degree in \(G\) among vertices in \(V(G)-S\). Then \(\alpha \ell < \delta' < 2\alpha \ell\).
Proof. Claim 2 guarantees \(\delta' > \alpha \ell\). We must prove \(\delta' < 2\alpha \ell\). Let \(v \in V(G)-S\) be an arbitrary vertex. Since \(d(v) > \alpha \ell \ge 2\ell > |S|\), \(v\) possesses at least one neighbor \(u \in V(G)-S\). Let \(e = uv\), and consider \(G_1 = G-e\). Because \(G\) is minimal, \(G_1\) must fail (i) or (ii).
If \(G_1\) fails (ii), \((G-e, S)\) contains a separation \((A,B)\) with \(|A\cap B|<|S|\). The edge \(e\) must cross this separation, forcing \(u \in A\setminus B\) and \(v \in B\setminus A\). Therefore, their common neighbors are restricted to \(A\cap B\). This implies \(|N(u)\cap N(v)| \le |A\cap B| < |S| \le \ell < \alpha \ell\), contradicting Claim 2.
Thus, \(G_1\) must fail (i), yielding \(\rho(V(G_1)-S) \le \lfloor\alpha \ell |V(G)-S|\rfloor - 1\). Since removing \(e\) drops the edge count in \(V(G)-S\) by precisely 1, we find exactly \(\rho(V(G)-S) \le \lfloor\alpha \ell |V(G)-S|\rfloor\).
The sum of the degrees of the vertices in \(V(G)-S\) satisfies: \[\begin{align} \sum_{w \in V(G)-S} d_G(w) &= 2|E(V(G)-S)| + |E(V(G)-S, S)| = 2\rho(V(G)-S) - |E(V(G)-S, S)|\\ &\le 2\lfloor\alpha \ell |V(G)-S|\rfloor -|E(V(G)-S, S)| < 2\lfloor\alpha \ell |V(G)-S|\rfloor\le 2\alpha \ell |V(G)-S|. \end{align}\] since \(|E(V(G)-S, S)|> 0\). By the Pigeonhole Principle, the minimum degree among these vertices, \(\delta'\), must be strictly less than \(2\alpha \ell\). ◻
Finally, select a vertex \(v\in V(G)-S\) with degree exactly \(\delta'\). Let \(H\) be the subgraph induced by \(v\) and its neighborhood \(N(v)\). The number of vertices in \(H\) is \(\delta'+1<1+ 2\alpha \ell\). By Claim 2, any \(u \in V(H)\) with \(u \notin S\) has \(d_H(u) = 1 + |N(u) \cap N(v)| > \alpha \ell\). For any \(u \in S_i\), its degree within \(H\) is strictly greater than \(\alpha \ell - 1 - |N(v) \cap S - N[u]| + 1 \ge \alpha \ell - |S_i|+1\), since \(u\) is adjacent all vertices in \(S-S_i\). This constructed subgraph \(H\) completely fulfills all requirements of the theorem. ◻
In this section, we show the dense graph \(H\) from Section 3 contains a knitted subgraph. This is a critical ingredient for a proof of our main result.
We will make use of the following result:
Theorem 7 (Liu, Rolek, and Yu [10]). Let \(G\) be a graph with \(n \ge 2\ell + 3\) vertices, where \(\ell \ge 5\). If \(\delta(G) \ge \frac{n+\ell}{2} - 1\), then \(G\) is \(\ell\)-knitted.
Theorem 8. Let \(\ell \ge 5\) be an integer, and let \(\alpha \ge 4\). Let \(H\) be a graph with \(|V(H)| \le \lceil 2\alpha \ell \rceil\), let \(T \subseteq V(H)\) with \(|T| \le \ell\). For any \(S \subseteq V(H)\) such that \(T \subseteq S\) and \(|S| = \ell\), and a partition of \(S=\cup_{i=1}^t S_i\) with \(s_i=|S_i|\), if \(v \in V(H)\), \[d(v)\ge \begin{cases} 1+\lfloor\alpha\ell\rfloor, & if v \in V(H) - T;\\ 1+\lfloor\alpha \ell\rfloor-s_i, & if v \in T\cap S_i. \end{cases}\] then \(H\) contains an \((s_1, \ldots, s_t)\)-knitted subgraph.
Proof of Theorem 8. Assume, for the sake of contradiction, that there is a subset \(S \subseteq V(H)\) with \(T \subseteq S\) and \(|S|=\ell\), and a partition \(S=\bigcup_{i=1}^{t} S_i\) such that \(H\) does not contain disjoint connected subgraphs corresponding to each \(S_i\). We analyze a partial \((\ell, t)\)-knit \(C=\bigcup_{i=1}^t C_i\), which is a subgraph of \(H\) wherein \(S_i\subseteq V(C_i)\) for all \(i\), but the components \(C_i\) are not strictly required to be connected.
We define an optimal \((\ell, t)\)-knit \(C=\bigcup_{i=1}^t C_i\) as a partial \((\ell, t)\)-knit that satisfies the following conditions in decreasing order of priority:
\(|V(C)|\le \min_{z\in S} d_H(z)\);
the total number of connected components of \(C\) is minimized;
subject to (a) and (b), \(|V(C)|\) is minimized.
By the optimality of \(C\), any component of \(C\) containing exactly one vertex of \(S\) must be an isolated vertex, and any component containing exactly two vertices of \(S\) must be a path connecting them. We may assume without loss of generality that \(S_1\subseteq V(C_1)\), but \(C_1\) is not connected. Thus, there exist \(x, y\in S_1\) belonging to different connected components of \(C_1\).
Since \(|V(C)|\le \min_{z\in S} d_H(z)\le d(x)\) and \(x\in C\), \(d(x, H-C)\ge 1\); in particular, \(H-C \neq \emptyset\).
Lemma 1. \(\delta^* \ge (\alpha - 1)\ell-c\), where \(c=|\{s_i\ge 2: 1\le i\le t and C_i is connected\}|\).
Proof. For every \(u\in V(H-C)\), we have \[d(u, H-C) = d(u,H) - d(u,C)\ge 1+\lfloor\alpha\ell\rfloor- d(u,C)\ge \alpha\ell-d(u,C).\]
We now bound \(d(u,C)\). Let \(C_i\) be the forest containing \(S_i\) in \(C\). Then if \(s_i=1\), the component \(C_i\) is a single vertex, so \(d(u, C_i) = 1 = s_i\); if \(s_i\ge 2\), Theorem 4 implies \(d(u, C_i)\le s_i+1\), with equality only if \(C_i\) is a tree and \(s_i\ge 2\). It follows that \[d(u, C) = \sum_{C_i} d(u,C_i) \le |S|+c=\ell+c.\] The lemma follows. ◻
Since \(x,y\) are not in the same component, so \(c\le \frac{\ell-2}{2}\); furthermore, for each \(S_i\subseteq S\) with \(x,y\not\in S_i\), \[c\le \frac{\ell-2-s_i}{2}+1=\frac{\ell-s_i}{2}.\]
Lemma 2. The subgraph \(H-C\) is connected.
Proof. Suppose that \(H-C\) has components \(H_1, \ldots, H_p\) where \(p\ge 2\). Since \(H_i\) does not inherently contain an \(\ell\)-knit, and \(|V(H_i)|\ge \delta^*\ge 2.5\ell\), by Lemma 1 and \(c\le \ell/2\) and \(\alpha\ge 4\). By Theorem 7, \(\delta^* < (|V(H_i)|+\ell)/2-1\). It follows that \[|V(H_i)| > 2\delta^*-\ell+2\ge (2\alpha-3)\ell-2c+2\ge (2\alpha-4)\ell+2.\] Since \(p \ge 2\), we sum the sizes: \(|V(H)| \ge |V(C)| + |V(H_1)| + |V(H_2)| > \ell + 2(2\alpha-4)\ell + 4\). Given \(|V(H)| < \lceil 2\alpha \ell\rceil\), this implies \(\lceil 2\alpha \ell\rceil > (4\alpha-7)\ell + 4\), which simplifies to \((2\alpha-7)\ell\le -3\). This contradicts our hypothesis that \(\alpha \ge 3.5\). ◻
Lemma 3. \(|V(C)| \ge \min_{z\in S} d_H(z)-4\).
Proof. Assume that \(|V(C)| \le \min_{z\in S} d_H(z)-5\). Let \(A=N(x)\cap V(H-C)\) and \(B=N(y)\cap V(H-C)\). Let \(A'=N(A)\cap V(H-C)-A\) and \(B'=N(B)\cap V(H-C)-B\). Finally, let \(D=V(H-C)-(A\cup A'\cup B\cup B')\). Since \(H-C\) is connected, there is a path from \(x\) to \(y\) through \(A\cup A'\cup D\cup B'\cup B\). If such a path has length at most \(6\), we could append this path to \(C\) to form a new knit \(C'\). This new knit \(C'\) would strictly reduce the number of connected components. Furthermore, its size would be bounded by \(|V(C')| \le |V(C)| + 5 \le \min_{z\in S} d_H(z)-5+5=\min_{z\in S} d_H(z)\). This directly contradicts the assumption that \(C\) is an optimal knit.
Since no such short path exists, take \(u\in D-N(A')\); \(u\) has no neighbors in \(A'\cup A\). Take \(v\in A\); the pair \(u, y\) share no common neighbors, nor do \(v, y\). Consequently, we can bound the size of \(H\): \[\begin{align} |V(H)| &\ge 3+d(y) + d(u, H-C) + d(v, H-C) \ge 3+\delta(H) + 2\delta^* \\ &\ge 3+\alpha\ell-s_i + (2\alpha - 2)\ell -2c= 3+(3\alpha - 2)\ell-s_i-2c\\ &\ge 3+(3\alpha - 2)\ell-s_i-2\cdot (\ell-s_i)/2=3+(3\alpha-3)\ell. \end{align}\] Since \(|V(H)|\le \lceil 2\alpha\ell\rceil\le 2\alpha\ell + 1\), this forces \(2\alpha\ell + 1 \ge 3+(3\alpha-3)\ell\), yielding \((\alpha-3)\ell\le -2\). This blatantly contradicts \(\alpha \ge 3\). ◻
Therefore, \(|V(C)| \ge \min_{z\in S} d_H(z)-4\). Then we have: \[|V(H-C)| = |V(H)| - |V(C)| \le \lceil 2\alpha \ell\rceil - \min_{z\in S} d_H(z)+4.\] Since \(|V(H-C)|\ge \delta^*+1\ge (\alpha-1)\ell-c+1\ge 2\ell+3\), we try to apply Theorem 7. Since \(H-C\) is not \(\ell\)-knitted, we have \[\begin{align} 0&>2\delta^* - \left(|V(H-C)| + \ell - 2 \right)\\ &\ge 2+2\lfloor\alpha\ell\rfloor-2d(u,C) - \big(\lceil 2\alpha \ell\rceil - \min_{z\in S} d_H(z)+4\big)- \ell + 2 \\ &= \min_{z\in S} d_H(z)-2d(u,C)+(2\lfloor\alpha\ell\rfloor-\lceil 2\alpha \ell\rceil-\ell)\\ &\ge (1+\lfloor\alpha\ell\rfloor-s_i)-2d(u,C)-\ell-(2\lfloor\alpha\ell\rfloor-\lceil 2\alpha \ell\rceil)\\ &\ge (1+\lfloor\alpha\ell\rfloor)-s_i-2(\ell+c)-\ell-(2\lfloor\alpha\ell\rfloor-\lceil 2\alpha \ell\rceil)\\ &\ge (1+\lfloor\alpha\ell\rfloor)-s_i-2\ell-2\cdot (\ell-s_i)/2-\ell-(2\lfloor\alpha\ell\rfloor-\lceil 2\alpha \ell\rceil)\\ &\ge (1+\lfloor\alpha\ell\rfloor)-4\ell-(2\lfloor\alpha\ell\rfloor-\lceil 2\alpha \ell\rceil)\\ &\ge (\alpha-4)\ell. \end{align}\] where \(d(u,H-C)=\delta^*\) for \(u\in H-C\), \(\min_{z\in S} d(z)=d(z)\ge 1+\lfloor\alpha\ell\rfloor-|S_i|\) for \(z\in S_i\). We used that \(d(u,C)=\sum_j d(u,C_j)\le \ell+c\), and \(c\le (\ell-|S_i|)/2\). A contradiction to \(\alpha\ge 4\). ◻
Combining Theorem 6 and Theorem 8, we deduce the following result, which encompasses Theorem 2.
Theorem 9. Let \(\ell\) be an integer. Let \(G\) be a graph and \(S\subseteq V(G)\) be a subset of size \(\ell\). If \((G,S)\) is \((4, \ell)\)-massed, then \((G,S)\) is knitted.
Proof. Assume, for the sake of contradiction, that there exists a \((4, \ell)\)-massed graph that is not \((s_1, \ldots, s_t)\)-knitted. We may select \(G\) such that \((G,S)\) is \((4,\ell)\)-minimal. By Theorems 6 and 8, \(G\) contains no rigid separation of order at most \(\ell\), but it strictly contains an \((s_1, \ldots, s_t)\)-knitted subgraph \(K\). Let \(S_1, \ldots, S_t\) with \(s_i=|S_i|\) be a partition of \(S\) that is not knitted.
By Menger’s Theorem, if there are \(\ell\) vertex-disjoint paths from \(S\) to \(K\), we can uniquely associate each vertex in \(S\) to a distinct terminal in \(K\) (take the first vertices on the paths that are on \(K\)). Take the corresponding partition of the terminals in \(K\) which can be connected via disjoint connected subgraphs inside \(K\). Tracing back through the paths resolves the knit in \(G\).
If no such \(\ell\) disjoint paths exist, there must be a separation \((A,B)\) separating \(S\) from \(K\) with \(S\subseteq A\) and \(K\subseteq B\), such that its order \(|A\cap B|\) is strictly less than \(\ell\). We choose \((A,B)\) to have the minimum possible order. By Menger’s Theorem again, there are \(|A\cap B|\) disjoint paths from \(A\cap B\) to \(K\). The identical routing logic demonstrates that for any partition of \(A\cap B\), the parts can be connected via \(K\). Thus, the induced subgraph \(G[B]\) securely knits \(A\cap B\). This perfectly satisfies Definition 3, meaning \((A,B)\) is a rigid separation of order at most \(\ell-1\), squarely contradicting the guarantee of Theorem 6. ◻
The authors would like to thank Michael Lafferty for the initial discussions that contributed to this revision.