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Claim. L. Chahua and J. Gutiérrez, On Tuza's conjecture in dense graphs, Discrete Appl. Math. 377 (2025), 225--233 (arXiv:2405.11409, v1 of 18 May 2024), prove three results on Tuza's inequality $\tau(G)\le2\nu(G)$, cited by the pages of arXiv v1:

• Theorem 5 (p. 3): "Let $G=(K,S,E(G))$ be a split graph on $n$ vertices. If $\delta(G)\ge\frac{3n}5$, then Conjecture 1 holds", where a split graph is one whose vertices split into a clique $K$ and an independent set $S$, and Conjecture 1 is Tuza's.
• Corollary 13 (p. 7), from Theorem 12 (p. 6): "For any $\alpha>0$, every tripartite graph $G$ with more than $\left(\frac{1+3\alpha}{12\alpha}\right)n^2$ edges satisfies $\tau(G)<\alpha\nu(G)$. In particular, if $G$ has more than $\frac{33n^2}{112}$ edges then $\tau(G)<\frac{28}{15}\nu(G)$." The abstract states the second sentence for tripartite graphs of minimum degree more than $\frac{33n}{56}$, which have more than $\frac{33n^2}{112}$ edges.
• Theorem 15 (p. 7): "For every complete 4-partite graph $G$ on at least five vertices, $\tau(G)\le\frac32\nu(G)$. Moreover, this bound it [sic] tight."

The three results are paged at Theorem 5, Corollary 13 and Theorem 15 of the library's source card.

Covers. Split graphs on $n$ vertices with minimum degree at least $3n/5$ (Theorem 5); complete $4$-partite graphs on at least five vertices, with $\tau\le\frac32\nu$ (Theorem 15); tripartite graphs on $n$ vertices with more than $\frac{33}{112}n^2$ edges, with $\tau<\frac{28}{15}\nu$ (Corollary 13). The statement for every graph stays open.

Depends on. Nothing in this wiki; the results are the paper's own theorems.

Acceptance. Refereed: Discrete Appl. Math. 377 (2025), 225--233, per its Crossref record. The page numbers cited are those of arXiv v1, not compared with the journal text. The site does not cite the paper.

Read depth. Claims checked: the three statements and the derivation of Corollary 13 from Theorem 12; the proofs of Theorems 5 and 15 are not checked, and that of Theorem 12 is checked for structure only.