Claims

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1969_03_01_brown_jung: Brown and Jung (1969) prove that a graph with chromatic number 5 and no K_5 contains two vertex-disjoint odd cycles, settling the instance a = b = 3 that Erdős asked; refereed in Acta Math. Acad. Sci. Hungar.

2008_12_28_balogh_kostochka_prince_stiebitz: Balogh, Kostochka, Prince and Stiebitz (2009) prove the Erdős–Lovász Tihany conjecture for quasi-line graphs and for graphs with independence number 2, for every k and every admissible pair; refereed in Discrete Math.

2018_05_27_song: Song (Discrete Math. 2019) proves that a graph with independence number at least 3 and no hole of length between 4 and twice that number minus 1 is (s,t)-splittable when omega < chi = s+t-1; refereed in Discrete Math.

2024_06_21_longbrake_tariq: Longbrake and Tariq (Discrete Math. 2026) prove the conjecture for pairs (s,t) with t at most s+2 when G contains K_s, with t at most 4s-3 when G contains K_s and is claw-free, and for (3,10) in claw-free graphs.

2026_07_22_song: Song (arXiv, 2026) proves that a graph with no induced C_4 whose every induced subgraph has a bisimplicial vertex is (s,t)-splittable when omega < chi = s+t-1, so the conjecture holds for even-hole-free graphs.

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