Geodesic
The chromaticity of complete split graphs
Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 208-221

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The join of null graph $O_m$ and complete graph $K_n$, $O_m+K_n=S(m,n)$, is called a complete split graph. In this paper, we characterize chromatically unique, determine list-chromatic number and characterize unique list colorability of the complete split graph $G=S(m,n)$. We shall prove that $G$ is chromatically unique if and only if $1\le m\le 2$, $ch(G)=n+1$, $G$ is uniquely $3$-list colorable graph if and only if $m\ge 4$, $n\ge 4$ and $m+n\ge 10$, $m(G)\le 4$ for every $1\le m\le 5$ and $n\ge 6$. Some the property of the graph $G=S(m,n)$ when it is $k$-list colorable graph also proved.
Keywords: chromatically unique, list- chromatic number, uniquely list colorable graph, complete split graph. 
@article{CHEB_2024_25_2_a11,
     author = {Hung Xuan Le},
     title = {The chromaticity of complete split graphs},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {208--221},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2024_25_2_a11/}
}
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Hung Xuan Le. The chromaticity of complete split graphs. Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 208-221. http://geodesic.mathdoc.fr/item/CHEB_2024_25_2_a11/