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On sublinear approximations for the Petersen coloring conjecture

04/19/2021
by   Davide Mattiolo, et al.
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If f:ℕ→ℕ is a function, then let us say that f is sublinear if lim_n→ +∞f(n)/n=0. If G=(V,E) is a cubic graph and c:E→{1,...,k} is a proper k-edge-coloring of G, then an edge e=uv of G is poor (rich) in c, if the edges incident to u and v are colored with three (five) colors. An edge is abnormal if it is neither rich nor poor. The Petersen coloring conjecture of Jaeger states that any bridgeless cubic graph admits a proper 5-edge-coloring c, such that there is no an abnormal edge of G with respect to c. For a proper 5-edge-coloring c of G, let N_G(c) be the set of abnormal edges of G with respect to c. In this paper we show that (a) The Petersen coloring conjecture is equivalent to the statement that there is a sublinear function f:ℕ→ℕ, such that all bridgeless cubic graphs admit a proper 5-edge-coloring c with |N_G(c)|≤ f(|V|); (b) for k=2,3,4, the statement that there is a sublinear function f:ℕ→ℕ, such that all (cyclically) k-edge-connected cubic graphs admit a proper 5-edge-coloring c with |N_G(c)|≤ f(|V|) is equivalent to the statement that all (cyclically) k-edge-connected cubic graphs admit a proper 5-edge-coloring c with |N_G(c)|≤ 2k+1.

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