Triangle-Covered Graphs in Extremal Theory

• Triangle-covered graphs are defined as graphs where every vertex belongs to at least one triangle, though related concepts also cover edge and clique perspectives.
• Research in this area establishes tight extremal bounds and explicit constructions, such as tree-of-bags and linear hypertree methods, to address triangle covering and packing problems.
• The study also tackles algorithmic challenges including NP-hard completion problems and approximations, connecting to broader themes like Tuza’s conjecture and clique covers.

Searching arXiv for relevant papers on triangle-covered graphs and related triangle covering/packing notions. Triangle-covered graph is a context-dependent term in graph theory rather than a single standardized definition. In recent edge-modification work, it denotes a graph in which every vertex belongs to at least one triangle (Madani et al., 14 Sep 2025). In older local covering problems, closely related papers study graphs in which every edge is contained in a triangle, or more generally in a copy of $K_k$ (Chakraborti et al., 2024, Chen et al., 2016). A different branch of the literature uses triangle cover for a set of edges or vertices meeting every triangle, and yet another uses triangle clique cover for collections of cliques that cover all $K_3$ subgraphs [(Yuster, 2010); (Liu et al., 2020); (Dau et al., 2017)]. This suggests that the subject is best organized by the underlying covering relation: local edge coverage, local vertex coverage, global triangle transversals, and clique-based coverings.

1. Definitions and notation

The edge-transversal formulation studies sets that hit every triangle. For a graph $G$, the triangle cover in the edge sense is

$$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$

and the triangle packing is

$$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$

Equivalent notation also appears as $\tau_t(G)$ for the minimum edge cover and $\nu_t(G)$ for the maximum packing [(Yuster, 2010); (Chen et al., 2016)].

A second, local notion concerns graphs in which every edge lies in a triangle. In (Chen et al., 2016), such graphs are called irreducible, meaning that every $e\in E$ lies in some triangle. In (Chakraborti et al., 2024), this is treated as the $k=3$ case of a $(k,1)$-cover, where every edge lies in a copy of $K_3$0.

A third notion is vertex-local. In (Madani et al., 14 Sep 2025), a graph $K_3$1 is triangle-covered if every vertex of $K_3$2 belongs to at least one triangle. The same paper defines a $K_3$3-completion set as a set of non-edges whose addition makes $K_3$4 triangle-covered.

Further variants are standard in adjacent literatures. The triangle covering number $K_3$5 may denote the minimum number of vertices that hit all triangles (Liu et al., 2020). A triangle clique cover is a set of cliques covering all triangles, with minimum size denoted $K_3$6 or $K_3$7 (Dau et al., 2017, Chen et al., 12 Jun 2025). The triangle-degree of a vertex is the number of triangles containing it (Falgas--Ravry et al., 2019). A plausible implication is that claims about “triangle-covered graphs” cannot be transferred across papers without checking whether the coverage condition is on edges, vertices, or the family of all triangles.

2. Edge-local coverage: every edge in a triangle

The classical starting point is a question of Erdős from 1988: what is the minimum number of edges in a connected $K_3$8-vertex graph where every edge is contained in a triangle? Catlin, Grossman, Hobbs, and Lai had already shown the stronger bound

$K_3$9

for graphs with $G$0 components (Chakraborti et al., 2024).

The exact modern formulation is for $G$1-covers. If $G$2 and $G$3 is a connected $G$4-vertex graph in which every edge lies in a copy of $G$5, write

$G$6

Then

$G$7

with equality if and only if $G$8, where $G$9 is the union of two $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$0s sharing $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$1 vertices (Chakraborti et al., 2024). The extremal family is described through a linear hypertree of cliques, or equivalently a tree-like gluing scheme in which overlaps are kept as small as possible.

For triangles, this specializes to

$$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$2

where $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$3 and $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$4 (Chakraborti et al., 2024). The extremal constructions are obtained by gluing triangles in a tree-like way. The same paper extends the problem to multiplicity constraints: for a connected graph in which every edge lies in at least two triangles, the sharp result is that the extremal graphs are precisely those with a $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$5-cover. It also shows that this coincidence fails in general: for $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$6, the minimum for a $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$7-cover is strictly less than for a $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$8-cover (Chakraborti et al., 2024).

This edge-local theory interacts directly with packing-covering questions. In (Chen et al., 2016), the irreducible condition is exactly the hypothesis needed for several algorithmic sufficient conditions toward Tuza’s conjecture.

3. Vertex-local coverage: every vertex in a triangle

Under the vertex-based definition, a triangle-covered graph is one in which every vertex belongs to at least one triangle (Madani et al., 14 Sep 2025). This is a genuinely different requirement from edge-local coverage: an edge may fail to lie in any triangle even though both its endpoints do.

For connected triangle-covered graphs of order $$T_t(G)=\min\{|S|: S\subseteq E(G),\ G-S\text{ is triangle-free}\},$$9, (Madani et al., 14 Sep 2025) gives a tight lower bound on the number of edges. If

$$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$0

then every connected triangle-covered graph $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$1 of order $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$2 satisfies

$$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$3

The bound is tight for every $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$4. The constructions are explicit: for $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$5, one uses $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$6 triangles connected by $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$7 edges in a tree-like fashion, while for $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$8 extra vertices are attached as specified in the paper. For $$v_t(G)=\max\{k:\text{there exist }k\text{ edge-disjoint triangles in }G\}.$$9, the extremal connected graphs admit a tree-of-bags decomposition in which each bag is either a triangle or one of finitely many exceptional graphs $\tau_t(G)$0 (Madani et al., 14 Sep 2025).

The same paper introduces the completion problem. A set

$\tau_t(G)$1

is a $\tau_t(G)$2-completion set if $\tau_t(G)$3 is triangle-covered. Deciding whether there exists a $\tau_t(G)$4-completion set of size at most $\tau_t(G)$5 is $\tau_t(G)$6-complete, does not admit a constant-factor approximation algorithm under standard complexity assumptions, and remains $\tau_t(G)$7-complete on connected bipartite graphs (Madani et al., 14 Sep 2025). On the positive side, the problem admits a polynomial-time $\tau_t(G)$8-approximation algorithm for general graphs via a Set-Cover reduction.

For special classes the situation is sharper. Every edge in a minimum $\tau_t(G)$9-completion set can be chosen between vertices at distance two. For trees, the minimum completion size ranges from $\nu_t(G)$0 to $\nu_t(G)$1, with paths attaining the lower bound and stars the upper bound, and there is a linear-time optimal algorithm (Madani et al., 14 Sep 2025). For chordal graphs,

$\nu_t(G)$2

where the $\nu_t(G)$3 arise from the decomposition into maximal tree components after deleting pendant clique vertices, again yielding a linear-time optimal algorithm. For $\nu_t(G)$4-edge-connected cactus graphs of order $\nu_t(G)$5,

$\nu_t(G)$6

In the random graph $\nu_t(G)$7, the threshold for triangle-coveredness is $\nu_t(G)$8 (Madani et al., 14 Sep 2025).

4. Global triangle transversals, packings, and Tuza-type inequalities

The most developed global notion of triangle coverage concerns edge sets meeting every triangle. Tuza’s conjecture asserts

$\nu_t(G)$9

for every graph $e\in E$0 (Chen et al., 2016). One extremal direction asks what happens when a graph is hard to make triangle-free. If $e\in E$1 has $e\in E$2 edges and density parameter $e\in E$3, then dense graphs satisfying

$e\in E$4

must contain a large triangle packing:

$e\in E$5

where $e\in E$6 is an absolute constant (Yuster, 2010). This improves the earlier $e\in E$7 lower bound that follows from the asymptotic validity of Tuza’s conjecture for dense graphs, and the same paper conjectures an asymptotically optimal packing size of $e\in E$8. The argument extends to larger cliques and odd cycles (Yuster, 2010).

For irreducible graphs, (Chen et al., 2016) provides polynomial-time sufficient conditions for Tuza’s conjecture. If any one of

$e\in E$9

holds, then a triangle cover of size at most $k=3$0 can be found in polynomial time. The proof passes through the triangle hypergraph and uses feedback sets in linear $k=3$1-uniform hypergraphs; in the second condition, a large bipartite subgraph supplies the cover directly (Chen et al., 2016).

Exact equality between packing and covering also occurs in specific structured families. In any bilaterally-complete tripartite graph, the maximum number of pairwise edge-disjoint triangles equals the minimum number of edges meeting all triangles:

$k=3$2

The proof combines Menger’s Theorem with König’s Line Colouring Theorem and generalizes the corresponding result for complete tripartite graphs due to Lakshmanan et al. (Amarnani et al., 2023). A different structural route appears in the theory of triangle graphs: if the triangle graph $k=3$3 is perfect, then Tuza’s conjecture holds for $k=3$4 (S. et al., 2014).

5. Triangle clique covers

Clique-cover variants ask not for edges or vertices that hit every triangle, but for cliques that contain every triangle. Formally, a $k=3$5 clique cover is a set of cliques such that every $k=3$6 of $k=3$7 is contained in one of them, and the minimum size is $k=3$8 (Dau et al., 2017). For $k=3$9, this is the triangle clique cover number $(k,1)$0.

The sharp extremal theorem is an exact triangle analogue of the Erdős–Goodman–Pósa theorem for edge clique covers. For every $(k,1)$1-vertex graph $(k,1)$2,

$(k,1)$3

with equality if and only if $(k,1)$4, the balanced complete tripartite Turán graph (Dau et al., 2017). Since

$(k,1)$5

this also yields the explicit bound

$(k,1)$6

again with equality only for $(k,1)$7 (Chen et al., 12 Jun 2025).

Algorithmically, the triangle clique cover problem is NP-hard in general; more generally, for every fixed $(k,1)$8, deciding whether $(k,1)$9 is NP-complete (Dau et al., 2017). On the positive side, the weighted $K_3$00 clique cover problem is solvable on semichordal graphs by extending the Scheinerman–Trenk framework (Dau et al., 2017). The broader Turán-type conjecture that $K_3$01 maximizes the $K_3$02-clique cover number was confirmed for $K_3$03 using inductive frameworks, greedy partition method, local adjustments, and clique-counting lemmas by Erdős and by Moon and Moser (Chen et al., 12 Jun 2025).

The clique-cover perspective is sometimes described informally as “triangle-covered,” but it is conceptually different from both local notions above: the object being covered is the family of triangles themselves, not the vertices or edges of the host graph.

6. Vertex-incidence thresholds and related structures

A related extremal problem asks for the edge density that forces vertex participation in triangles. In (Falgas--Ravry et al., 2019), the triangle-degree of a vertex is the number of triangles containing it, and the function $K_3$04 measures the asymptotically smallest possible maximum scaled triangle-degree in an $K_3$05-vertex graph of edge density at least $K_3$06. The paper gives explicit piecewise upper bounds for $K_3$07, conjectures these bounds are tight for all $K_3$08, proves the conjecture for tripartite graphs, and uses flag algebra computations to obtain nearly sharp lower bounds on $K_3$09 (Falgas--Ravry et al., 2019). The associated covering threshold is the smallest $K_3$10 such that $K_3$11, and the conjectured sharp value is

$K_3$12

For tripartite graphs, if $K_3$13, then

$K_3$14

These bounds are sharp in the tripartite setting (Falgas--Ravry et al., 2019).

The phrase triangle covering number is also ambiguous on the transversal side. In (Liu et al., 2020), $K_3$15 denotes the minimum number of vertices that hit all triangles. For large $K_3$16 and $K_3$17, an $K_3$18-vertex graph with $K_3$19 edges and $K_3$20 satisfies a sharp lower bound

$K_3$21

with equality only for the constructions $K_3$22 or $K_3$23 except for a few small exceptional cases (Liu et al., 2020). This extends the Erdős–Rademacher line of results beyond the classical regime $K_3$24.

Several adjacent constructions should not be conflated with triangle-coveredness itself. The triangle graph $K_3$25 has one vertex for each triangle of $K_3$26, with adjacency defined by shared edges; its cycle, tree, chordal, and perfect cases admit forbidden-subgraph characterizations (S. et al., 2014). A triangle-forest is a graph with no cycles of length more than three, and every planar graph can be partitioned into two induced triangle-forests; this was already proved by Carsten Thomassen in 1995 (Knauer et al., 2024). In algorithmic triangle counting, a cover-edge set is a subset of edges containing at least one edge from every triangle; BFS level structure shows that the set of horizontal edges forms such a cover-edge set, enabling sequential, shared-memory, and distributed triangle-counting algorithms (Bader et al., 2024). These notions are structurally adjacent because they organize triangles globally, but they solve different problems.

The modern literature therefore treats “triangle-covered graph” less as a single invariant than as a family of covering paradigms. The edge-local and vertex-local versions lead to different extremal constructions and different completion problems; transversal formulations connect to Tuza’s conjecture and dense graph packing; clique-cover variants recover Turán-type extremal bounds; and vertex-incidence thresholds bring in triangle-degree, flag algebras, and generalized Erdős–Rademacher theory.