WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 25, 2026
The Complete b-chromatic sum of the Mycielskian of Paths
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Abstract: A b-coloring of a graph $$ G $$ is a proper vertex-coloring that for each class of color $$ i, $$ there is a vertex whose neighbors are of all colors but $$ i. $$ The b-chromatic number of $$ G $$ is the largest positive integer $$ \kappa $$ where a b-coloring of $$ \kappa $$ colors exists. The b-chromatic sum of $$ G $$ is the minimum sum of the colors of all vertices of $$ G $$ over all possible b-colorings that give the b-chromatic number. For the Mycielskian of path $$ \mu(P_n), $$ the b-chromatic sum $$ \varphi'\left(\mu(P_n)\right) $$ is known for all $$ n $$ except when $$ 10\leq n\leq 15. $$ In this work, we give the value of $$ \varphi'\left(\mu(P_n)\right) $$ when $$ 10\leq n\leq 15. $$ This completes the list of the b-chromatic sum of $$ \mu(P_n)$$.
Keywords:
b-coloring, b-domination, b-chromatic number, b-chromatic sum, The Minimum Sum, Mycielskian, path
Pages: 197-201
DOI: 10.37394/23206.2026.25.19