@article{Yuan2026, 
author = {Yufan Yuan},
title = {Progress on the Borodin–Kostochka conjecture: A structural approach via vertex partitions relative to a maximum clique},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {8492-8506},
keywords = {Borodin-Kostochka conjecture, chromatic number, forbidden induced graphs, K1, t--free},
url = {https://www.sciopen.com/article/10.3934/math.2026349},
doi = {10.3934/math.2026349},
abstract = {The Borodin-Kostochka conjecture states that for any graph    G with    Δ  (  G  )  ≥  9, we have    χ  (  G  )  ≤  max  {  Δ  (  G  )  −  1  ,  ω  (  G  )  }. In this paper, we study the structure of potential counterexamples by partitioning vertices according to the number of neighbors they have in a fixed maximum clique. This approach provides a sufficient condition for    χ  (  G  )  ≤  Δ  (  G  )  −  1. Consequently, we confirm the conjecture for any              K              1        ,        t              ¯  -free graph    G with    t  ≥  3 and    Δ  (  G  )  ≥  2  t  +  1, strengthening and extending the recent work of Lan and Lin in 2024.}
}