Size conditions and spectral conditions for generalized factor-critical (bicritical) graphs and $k$-$d$-critical graphs

Let $\mbox{odd}(G)$ and $i(G)$ denote the number of nontrivial odd components and the number of isolated vertices of a graph $G$, respectively. The $k$-Berge-Tutte-formula of a graph $G$ is defined as: $\mbox{def}k(G)=\mathop{\text{max}}\limits{S\subseteq V(G)}{k\cdot i(G-S)-k|S|} $ for even $k$; $\mbox{def}k(G)=\mathop{\mbox{max}}\limits{S\subseteq V(G)}{\mbox{odd}(G-S)+k\cdot i(G-S)-k|S|} $ for odd $k$. A $k$-barrier of a graph $G$ is the subset $S\subseteq V(G)$ that reaches the maximum value in the $k$-Berge-Tutte-formula of $G$. A graph $G$ of odd order (resp. even order) is generalized factor-critical (resp. generalized bicritical) if $\emptyset$ is its only $k$-barrier. Denote by $E_G(v)$ the set of all edges incident to a vertex $v$ in $G$. A $k$-matching of a graph $G$ is a function $f:E(G) \rightarrow {0,1,…,k}$ such that $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for every vertex $v\in V(G)$. For $1\leq d\leq k$ and $d \equiv |V(G)|$(mod 2), if for any $ v \in V(G)$, there exists a $k$-matching $f$ such that $\sum_{e\in E_G(v)}f(e)=k-d$ and $\sum_{e\in E_G(u)}f(e)=k \text{ for any } u\in V(G)-{v}$. Then $G$ is $k$-$d$-critical. In this paper, we establish tight sufficient conditions in terms of size or spectral radius respectively for a graph $G$ to be generalized factor-critical, generalized bicritical, and $k$-$d$-critical. Furthermore, we prove the equivalence of the existence of four factors (namely, ${K_2,{C_t: t\geq 3}}$-factor, ${K_2,{C_{2t+1}:t\geq 1 }}$-factor, fractional perfect matching, perfect $k$-matching with even $k$) in a graph. Thus we also give size conditions and spectral radius conditions for a graph $G-v$ to have one of the four factors for any $v\in V(G)$.

💡 Research Summary

The paper investigates three families of graphs—generalized factor‑critical (GFCₖ) graphs, generalized bicritical (GBCₖ) graphs, and k‑d‑critical graphs—by establishing tight sufficient conditions expressed in terms of either the number of edges (size) or the spectral radius of the adjacency matrix. The authors work with the k‑Berge‑Tutte formula, a natural extension of the classical Berge‑Tutte theorem to k‑matchings. For even k the deficiency is defined as
 defₖ(G)=max_{S⊆V(G)}{k·i(G−S)−k|S|},
and for odd k as
 defₖ(G)=max_{S⊆V(G)}{odd(G−S)+k·i(G−S)−k|S|},
where i(·) counts isolated vertices and odd(·) counts non‑trivial odd components. A set S attaining the maximum is called a k‑barrier; a graph of odd order is GFCₖ if the empty set is its only k‑barrier, while a graph of even order is GBCₖ under the same condition.

Using Lemma 2.3 the authors translate the barrier condition into simple inequalities: for even k, GFCₖ (resp. GBCₖ) is equivalent to i(G−S)≤|S|−1 for all non‑empty S; for odd k, GFCₖ requires odd(G−S)+k·i(G−S)≤k|S|−1, and GBCₖ requires the same inequality with “−2” on the right‑hand side. Lemma 2.4 gives the analogous condition for k‑d‑critical graphs: odd(G−S)+k·i(G−S)≤k|S|−d for all non‑empty S, where d≡|V(G)| (mod 2).

Size conditions.
Theorem 3.1 (odd k≥3) shows that for a graph of odd order n≥7, if the number of edges satisfies
 e(G) ≥ C(n−1,2)+1,
then G is GFCₖ, with the unique extremal counterexample K₁ ∨ (K_{n−2}+K₁). For the small orders n=5 and n=3 the authors list the corresponding exceptional graphs (K₁ ∨ (K₃+K₁) and K₂ ∨ 3K₁). The proof proceeds by assuming a non‑GFCₖ graph with the maximum possible number of edges, then applying Lemma 2.5 (a graph‑joining inequality) to show that such a graph must be of the extremal form, contradicting the edge‑count hypothesis.

Theorem 4.1 (odd k≥3, even order) gives analogous results for GBCₖ. For n≥10, the same edge bound e(G) ≥ C(n−1,2)+1 forces G to be GBCₖ, again with K₁ ∨ (K_{n−2}+K₁) as the only extremal obstruction. For the smaller even orders 4≤n≤8 the paper provides tighter bounds (e.g., e(G)≥3n²−2n/8 for n=4,6 and e(G)≥22 for n=8) and lists the corresponding exceptional graphs (K_{n/2} ∨ (n/2)K₁, K₄ ∨ 4K₁, etc.). The proof mirrors that of Theorem 3.1, exploiting the maximal‑edge assumption and the structure forced by Lemma 2.5.

For k‑d‑critical graphs, Lemma 2.4 yields the necessary inequality; the authors then derive explicit edge‑count thresholds that guarantee the inequality holds for all S, thereby ensuring the graph is k‑d‑critical. The thresholds depend linearly on d: larger d relaxes the required density.

Spectral radius conditions.
Theorem 3.2 states that if the spectral radius satisfies
 ρ(G) ≥ ρ(K₁ ∨ (K_{n−2}+K₁)),
then G is GFCₖ (again, except for the extremal graph itself). The proof uses Lemma 2.1, which bounds the spectral radius by √(2m−n+1), to translate the edge‑count condition of Theorem 3.1 into a spectral condition. By computing ρ(K₁ ∨ (K_{n−2}+K₁)) = n−2, the authors show that any graph with a larger spectral radius must have at least the required number of edges, and therefore must be GFCₖ.

Theorem 4.2 gives the analogous spectral condition for GBCₖ, with the same lower bound ρ(K₁ ∨ (K_{n−2}+K₁)). The argument is identical: a larger spectral radius forces enough edges to satisfy the size condition of Theorem 4.1, and the extremal graph is the only case where equality holds.

Equivalence of four factor concepts.
Section 6 proves that the following four properties are equivalent for any graph:

1. Existence of a {K₂, {C_t : t≥3}}‑factor (i.e., a spanning subgraph whose components are either edges or cycles of length at least three).
2. Existence of a {K₂, {C_{2t+1} : t≥1}}‑factor (spanning subgraph with edges and odd cycles).
3. Existence of a fractional perfect matching (µ_f(G)=|V(G)|/2).
4. Existence of a perfect k‑matching for any even k (µ_k(G)=k·|V(G)|/2).

The proof proceeds by showing that each condition can be transformed into the next using Lemma 2.2 (which guarantees a fractional matching can be taken with values 0, ½, 1) and the k‑Berge‑Tutte inequalities. In particular, a fractional perfect matching can be “scaled” to a perfect k‑matching when k is even, and a perfect k‑matching can be decomposed into a collection of edges and cycles, yielding the two factor families.

Consequences for vertex‑deleted subgraphs.
Because of the equivalence, the authors obtain size and spectral conditions guaranteeing that for every vertex v, the graph G−v possesses at least one of the four factors. Concretely, if G satisfies the edge‑count or spectral‑radius thresholds established for GFCₖ (or GBCₖ), then removing any vertex leaves a graph that still meets the corresponding factor condition. This extends classical results on the robustness of perfect matchings to a broader class of spanning substructures.

Overall significance.
The paper unifies three strands of graph theory—matching theory, factor theory, and spectral graph theory—by providing sharp, easily computable criteria for complex structural properties. The results are tight: the authors identify the exact extremal graphs where the inequalities become equalities, proving that no stronger universal bound can exist. Moreover, the equivalence of the four factor concepts bridges combinatorial and linear‑programming perspectives (fractional matchings) and shows that spectral information alone can certify the existence of rich spanning substructures. These contributions are valuable both for theoretical investigations (e.g., characterizing graph classes with prescribed matching/ factor properties) and for practical applications such as network design, where edge density or spectral radius are often more readily observable than detailed component counts.