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 $\chi(G-e) = \chi(G) - 1$.
• $P$0 is edge-critical (for chromatic number) if every $P$1 is critical, i.e., removing any edge reduces $P$2 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 $P$3, but in an edge-critical graph, every such removal is maximally disruptive (Kaiser et al., 2019, Paulusma et al., 2017).

Other Edge-Critical Notions

• Edge-$P$4-critical (edge-chromatic-critical) graphs: $P$5 with maximum degree $P$6, chromatic index $P$7, and $P$8 for every $P$9. This type of edge-criticality underpins much of edge-coloring theory (Vizing, Goldberg) (Cao et al., 2017, Cao 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 $G$0-colorable graphs: Uniquely $G$1-colorable graphs $G$2 such that $G$3 is not uniquely $G$4-colorable for every $G$5 (Li et al., 2013).

A formal equivalence of edge deletion and edge contraction appears in chromatic edge-criticality: for any graph $G$6 and $G$7, $G$8 is critical if and only if contracting $G$9 reduces the chromatic number by 1 (Paulusma et al., 2017).

2. Structural Results and Extremal Theorems

Uniquely $P$0-Colorable Planar Graphs

An edge-critical uniquely $P$1-colorable graph is one where the chromatic number is $P$2, there is a unique $P$3-coloring (up to permutation), and removal of any edge breaks both uniqueness and $P$4-colorability (Li et al., 2013):

• For planar, uniquely 3-colorable, edge-critical graphs, the sharp size bound is $P$5 for $P$6.
• The extremal constructions combine outerplanar triangle chains with sparse interconnection, enforcing the criticality and uniqueness restriction.

Extremal examples exist meeting $P$7 for $P$8, 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 $P$9 and their Schrijver subgraphs $e\in E$0, edge-criticality takes the form: no proper subgraph maintains the chromatic number. For $e\in E$1, a family $e\in E$2 is constructed with $e\in E$3 and the property that $e\in E$4 for each edge $e\in E$5 (Kaiser et al., 2019).

Edge-$e\in E$6-Critical Graphs in Edge Coloring

Vizing's and Goldberg's theories focus on class II graphs (where $e\in E$7):

• Every edge of an edge-$e\in E$8-critical $e\in E$9 is critical; $G-e$0 admits a proper $G-e$1-coloring (Cao et al., 2017).
• For $G-e$2 large, such graphs have tightly controlled average degree; the current best lower bound is

$G-e$3

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

In the subcubic case ($G-e$4), the critical mean degree is $G-e$5, sharp for the o-join of two $G-e$6s (Petersen graph minus a vertex). This provides the most restrictive possible density for 3-critical graphs other than $G-e$7 (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 $G-e$8 edge-critical with chromatic number $G-e$9 ($P$0), maximizing the size of $P$1-vertex $P$2-free graphs leads uniquely (for large $P$3) to the balanced $P$4-partite graph $P$5 (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

$P$6

where $P$7 denotes the maximal number of edges in a graph of matching and degree at most $P$8 (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 $P$9 (the forbidden induced subgraph) is contained in $\chi(G)$0 or $\chi(G)$1 (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 $\chi(G)$2-free graphs, mirroring the Lovász-Král-Kratochvíl dichotomy for coloring.

In cases where the recognition is tractable (cographs and $\chi(G)$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 $\chi(G)$4-connected factor-critical, a $\chi(G)$5-connected bipartite ECE-graph, or an even clique.
• Factor-critical ECE-graphs with connectivity $\chi(G)$6 are precisely classified via five structural types according to Favaron's theory.
• For connectivity $\chi(G)$7, ECE-graphs are characterized by $\chi(G)$8, maximal triangle-free complement, and the nonexistence of dominating edges.

Vertex-critical equimatchable (VCE) graphs are a related family: $\chi(G)$9 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 $e\in E(G)$0 with $e\in E(G)$1, every $e\in E(G)$2-vertex $e\in E(G)$3-free graph with $e\in E(G)$4 can be converted into $e\in E(G)$5 by at most $e\in E(G)$6 edge additions/removals (Roberts et al., 2016).
• The threshold $e\in E(G)$7 and the geometric mean bound $e\in E(G)$8 both reflect sharp stability above classical Erdős-Simonovits $e\in E(G)$9 statements.

In the suspension context, the extremal graphs are precisely those obtained from $\chi(G-e) = \chi(G) - 1$0 with optimal insertion of graphs of degree and matching at most $\chi(G-e) = \chi(G) - 1$1 in one part (Hou et al., 2022).

The stability and extremal number theorems rest essentially on $\chi(G-e) = \chi(G) - 1$2 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 $\chi(G-e) = \chi(G) - 1$3 in Schrijver graphs (Kaiser et al., 2019)).
• Improving density lower bounds for edge-$\chi(G-e) = \chi(G) - 1$4-critical graphs (e.g., approaching Vizing's conjectured $\chi(G-e) = \chi(G) - 1$5 (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.