@article{Zhou2022, 
author = {Jinqiu Zhou and Qunfang Li},
title = {A note on PM-compact        K    4  -free bricks},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {3},
pages = {3648-3652},
keywords = {matching covered graph, perfect matching and brick},
url = {https://www.sciopen.com/article/10.3934/math.2022201},
doi = {10.3934/math.2022201},
abstract = {A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching covered graphs. Lovász (Combinatorica, 3 (1983), 105-117) showed that every brick is        K    4  -based or              C      ¯        6  -based. A brick is        K    4  -free (respectively,              C      ¯        6  -free) if it is not        K    4  -based (respectively,              C      ¯        6  -based). Recently, Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 34(3) (2020), 1769-1790) characterised the PM-compact              C      ¯        6  -free bricks. In this note, we show that, by using the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769-817), the only PM-compact        K    4  -free brick is              C      ¯        6  , up to multiple edges.}
}