On total and edge coloring some Kneser graphs

C. M. H. de Figueiredo, C. S. R. Patrão, D. Sasaki () and M. Valencia-Pabon
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C. M. H. de Figueiredo: COPPE, Universidade Federal do Rio de Janeiro
C. S. R. Patrão: COPPE, Universidade Federal do Rio de Janeiro
D. Sasaki: IME, Universidade do Estado do Rio de Janeiro
M. Valencia-Pabon: LIPN, Université Sorbonne Paris Nord

Journal of Combinatorial Optimization, 2022, vol. 44, issue 1, No 6, 119-135

Abstract: Abstract In this work, we investigate the total and edge colorings of the Kneser graphs K(n, s). We prove that the sparse case of Kneser graphs, the odd graphs $$O_k=K(2k-1,k-1)$$ O k = K ( 2 k - 1 , k - 1 ) , have total chromatic number equal to $$\Delta (O_k) + 1$$ Δ ( O k ) + 1 . We prove that Kneser graphs K(n, 2) verify the Total Coloring Conjecture when n is even, or when n is odd not divisible by 3. For the remaining cases when n is odd and divisible by 3, we obtain a total coloring of K(n, 2) with $$\Delta (K(n,2)) + 3$$ Δ ( K ( n , 2 ) ) + 3 colors when $$n \equiv 3~\hbox {mod}~4$$ n ≡ 3 mod 4 , and with $$\Delta (K(n,2)) + 4$$ Δ ( K ( n , 2 ) ) + 4 colors when $$n \equiv 1~\hbox {mod}~4$$ n ≡ 1 mod 4 . Furthermore, we present an infinite family of Kneser graphs K(n, 2) that have chromatic index equal to $$\Delta (K(n,2))$$ Δ ( K ( n , 2 ) ) .

Keywords: Total coloring; Edge coloring; Odd graphs; Kneser graphs; 05C15; 05C85; 05C69; 05C76 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-021-00816-z

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