Edge-Critical Graphs

• Edge-Critical Graphs are defined as graphs in which every edge is critical, meaning its removal reduces a specified property such as the chromatic number.
• They underpin sharp extremal constructions and stability theorems in graph theory, offering tight bounds in coloring, matching, and edge-coloring frameworks.
• Their study bridges structural, algorithmic, and complexity theory, with practical applications in graph coloring, equimatchability, and NP-hard recognition problems.

An edge-critical graph is a graph in which the removal of any edge strictly reduces a specified critical graph property, typically its chromatic number or another critical property (e.g., equimatchability or edge-chromatic index). Edge-criticality appears in various guises throughout structural, extremal, and algorithmic graph theory, often forming the foundation for extremal constructions, sharp stability theorems, and complexity dichotomies. The literature encompasses edge-criticality for vertex colorings, edge colorings, matchings, and structural subgraph properties.

1. Definitions and General Principles

An undirected, simple graph $G=(V,E)$ is edge-critical with respect to a property $P$ (e.g., coloring, matching) if $G$ satisfies $P$, but for every edge $e\in E$, the graph $G-e$ fails to satisfy $P$.

Vertex-Coloring Edge-Criticality

The most classical context for edge-criticality considers the chromatic number $\chi(G)$:

• An edge $e\in E(G)$ is critical if $\chi(G-e) = \chi(G) - 1$.
• $G$ is edge-critical (for chromatic number) if every $e\in E(G)$ is critical, i.e., removing any edge reduces $\chi$ by exactly 1 (Paulusma et al., 2017).

In various research, generalizations compare edge- to vertex-criticality (a graph is vertex-critical if the deletion of any vertex lowers its chromatic number), noting that edge-criticality is a strictly stronger (sparser) notion: in a vertex-critical graph, edge removals may have no effect on $\chi$, but in an edge-critical graph, every such removal is maximally disruptive (Kaiser et al., 2019Paulusma et al., 2017).

Other Edge-Critical Notions

• Edge-$\Delta$-critical (edge-chromatic-critical) graphs: $G$ with maximum degree $\Delta(G)=\Delta$, chromatic index $\chi'(G) = \Delta+1$, and $\chi'(G-e) = \Delta$ for every $e\in E(G)$. This type of edge-criticality underpins much of edge-coloring theory (Vizing, Goldberg) (Cao et al., 2017Cao et al., 2017).
• Edge-critical equimatchable graphs (ECE-graphs): Equimatchable graphs in which removal of any edge destroys equimatchability (Deniz et al., 2022).
• Edge-critical uniquely $k$-colorable graphs: Uniquely $k$-colorable graphs $G$ such that $G-e$ is not uniquely $k$-colorable for every $e$ (Li et al., 2013).

A formal equivalence of edge deletion and edge contraction appears in chromatic edge-criticality: for any graph $G$ and $e\in E(G)$, $e$ is critical if and only if contracting $e$ reduces the chromatic number by 1 (Paulusma et al., 2017).

2. Structural Results and Extremal Theorems

Uniquely $k$-Colorable Planar Graphs

An edge-critical uniquely $k$-colorable graph is one where the chromatic number is $k$, there is a unique $k$-coloring (up to permutation), and removal of any edge breaks both uniqueness and $k$-colorability (Li et al., 2013):

• For planar, uniquely 3-colorable, edge-critical graphs, the sharp size bound is $|E(G)| \leq \frac{5}{2}n - 6$ for $n\geq6$.
• The extremal constructions combine outerplanar triangle chains with sparse interconnection, enforcing the criticality and uniqueness restriction.

Extremal examples exist meeting $|E(G)| = \frac{5}{2}n-7$ for $n=10,12,14$, constructed from chains of triangles plus pendant vertices (Li et al., 2013).

Edge-Critical Subgraphs of Kneser and Schrijver Graphs

In the context of Kneser graphs $KG(n,k)$ and their Schrijver subgraphs $SG(n,k)$, edge-criticality takes the form: no proper subgraph maintains the chromatic number. For $k=2$, a family $H_n$ is constructed with $\chi(H_n)=n-2$ and the property that $\chi(H_n-e) = n-3$ for each edge $e$ (Kaiser et al., 2019).

Edge-$\Delta$-Critical Graphs in Edge Coloring

Vizing's and Goldberg's theories focus on class II graphs (where $\chi'(G)=\Delta+1$):

• Every edge of an edge-$\Delta$-critical $G$ is critical; $G-e$ admits a proper $\Delta$-coloring (Cao et al., 2017).
• For $\Delta$ large, such graphs have tightly controlled average degree; the current best lower bound is

$$\bar{d}(G) \geq \begin{cases} 0.69241\,\Delta - 0.15658, & \Delta\geq 66, \ 0.69392\,\Delta - 0.20642, & \Delta=65, \ 0.68706\,\Delta + 0.19815, & 56\leq \Delta\leq64. \end{cases}$$

(Cao et al., 2017), strictly improving earlier bounds.

In the subcubic case ($\Delta=3$), the critical mean degree is $\frac{46}{17}$, sharp for the o-join of two $P^*$s (Petersen graph minus a vertex). This provides the most restrictive possible density for 3-critical graphs other than $P^*$ (Cranston et al., 2015).

Edge-Criticality in Extremal Graph Theory

Edge-critical graphs play a fundamental role in extremal Turán-type constructions and stability theorems:

• For $H$ edge-critical with chromatic number $r$ ($\chi(H)=r$), maximizing the size of $n$-vertex $H$-free graphs leads uniquely (for large $n$) to the balanced $(r-1)$-partite graph $T_{r-1}(n)$ (Roberts et al., 2016).
• When considering suspensions of edge-critical graphs---that is, adding a universal vertex to a multiset of such graphs---the extremal number is determined by

$\textrm{ex}(n,H) = t_{r-1}(n) + f(k-1,k-1)$

where $f(k-1,k-1)$ denotes the maximal number of edges in a graph of matching and degree at most $k-1$ (Hou et al., 2022).

3. Algorithmic and Complexity Aspects

Determining the existence of an edge whose removal reduces chromatic number by one is polynomial-time solvable if and only if $H$ (the forbidden induced subgraph) is contained in $P_4$ or $P_1+P_3$ (i.e., the class is perfect/cograph or disjoint union of a vertex and path), and otherwise is NP-hard or coNP-hard (Paulusma et al., 2017). This establishes a sharp complexity dichotomy for edge-critical recognition in $H$-free graphs, mirroring the Lovász-Král-Kratochvíl dichotomy for coloring.

In cases where the recognition is tractable (cographs and $(P_1+P_3)$-free), explicit certifying algorithms exist; in the hard cases (e.g., claw-free, cycle-free), the problem is provably intractable unless P=NP (Paulusma et al., 2017).

4. Edge-Criticality in Matchings and Equimatchable Graphs

Edge-critical equimatchable graphs (ECE-graphs) are defined as equimatchable graphs for which every edge is critical, i.e., removal of any edge destroys equimatchability (Deniz et al., 2022). Their structure is sharply constrained:

• Every ECE-graph is either $2$-connected factor-critical, a $2$-connected bipartite ECE-graph, or an even clique.
• Factor-critical ECE-graphs with connectivity $2$ are precisely classified via five structural types according to Favaron's theory.
• For connectivity $k\geq3$, ECE-graphs are characterized by $\alpha(G)=2$, maximal triangle-free complement, and the nonexistence of dominating edges.

Vertex-critical equimatchable (VCE) graphs are a related family: $G$ is equimatchable and the removal of any vertex destroys equimatchability. Every factor-critical ECE-graph is VCE, but the converse does not hold; bipartite ECE-graphs are disjoint from VCE graphs.

Additionally, there is a direct correspondence between ECE-graphs and well-covered line-graphs without shedding vertices, answering a prominent open problem (Deniz et al., 2022).

5. Stability and Extremal Applications

Edge-critical graphs (for the chromatic number) have maximal impact in extremal stability contexts:

• For edge-critical $H$ with $\chi(H) = k+1$, every $n$-vertex $H$-free graph with $e(G) \geq t_k(n) - f(n)$ can be converted into $T_k(n)$ by at most $C_H f(n)^{1/2} n$ edge additions/removals (Roberts et al., 2016).
• The threshold $f(n) = o(n^2)$ and the geometric mean bound $f(n)^{1/2} n$ both reflect sharp stability above classical Erdős-Simonovits $o(n^2)$ statements.

In the suspension context, the extremal graphs are precisely those obtained from $T_{r-1}(n)$ with optimal insertion of graphs of degree and matching at most $k-1$ in one part (Hou et al., 2022).

The stability and extremal number theorems rest essentially on $H$ being edge-critical; without this, the Turán graph is not uniquely extremal, and asymptotic stability cannot be guaranteed.

6. Open Problems and Further Directions

Open questions include:

• Determining sharp extremal families in more general classes (e.g., higher $k$ in Schrijver graphs (Kaiser et al., 2019)).
• Improving density lower bounds for edge-$\Delta$-critical graphs (e.g., approaching Vizing's conjectured $\bar{d}(G) \geq \Delta-1+2$ (Cao et al., 2017)).
• Algorithmic classification of edge-critical recognition for other hereditary properties or in parameterized complexity frameworks (Paulusma et al., 2017).
• Full characterization of edge-critical equimatchable graphs with higher connectivity and lower degree conditions (Deniz et al., 2022).

These directions highlight the centrality and technical richness of edge-criticality across structural, extremal, and computational graph theory.