---

Claim. Theorem 2 (p. 108) of the paper states: "Let $F_1=K_{1,n_1}\cup K_{1,n_2}\cup\cdots\cup K_{1,n_s}$ and $F_2=K_{1,m_1}\cup K_{1,m_2}\cup\cdots\cup K_{1,m_t}$ with $n_1\ge n_2\ge\cdots\ge n_s$ and $m_1\ge m_2\ge\cdots\ge m_t$. Set $\ell_k=\max\{n_i+m_j-1:i+j=k\}$ for all $2\le k\le s+t$. If

$$\binom{\ell_j}2>\sum_{i=j}^{s+t}\ell_i\quad\text{for all }2\le j\le s+t,$$

then $\hat r(F_1,F_2)=\sum_{k=2}^{s+t}\ell_k$." The paper's $\ell_k$ is the $l_k$ of Problem 561, so this is the conjectured formula under the stated condition. The inequality is strict as printed, here and in the announcement on p. 106. Since the sum on the right contains $\ell_j$ itself and $\binom{\ell}2>\ell$ needs $\ell\ge4$, the hypothesis forces every $\ell_k\ge4$, and so excludes every pair with $n_s+m_t\le4$. The proof (pp. 108--109) shows that a minimal arrowing graph contains the stars $K_{1,\ell_k}$ edge-disjointly, using Vizing's theorem and the paper's Theorem 1 (p. 106) on red-blue colorings with bounded degree in both colors. The paper says (p. 109) that Theorem 2 does not establish the conjecture in general. The theorem is paged as Theorem 2 of the library's source card.

Covers. The formula for every pair of star forests with $\binom{l_k}2>\sum_{i=k}^{s+t}l_i$ for every $2\le k\le s+t$. The formula for all star forests is not claimed.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. Refereed: E. Győri and R. H. Schelp, Two-edge colorings of graphs with bounded degree in both colors, Discrete Math. 249 (2002), no. 1--3, 105--110 (received 29 June 1999, accepted 26 March 2001). The Crossref record dates the issue April 2002, the month this page is dated by; the day is a placeholder. The site's commentary credits the condition to this paper, but the site labels the problem OPEN, so its pages are not acceptance.

Read depth. The statement, its announcement on p. 106 and the proof of Theorem 2 were read and the proof's reduction followed; the proof of Theorem 1 was read for structure only. Nothing is independently reviewed in this corpus.