In 1968, Vizing proposed the following conjecture: If G = (V, E) is a ∆-critical graph of order n and size m, then m ≥ 1 2 [(∆ − 1)n + 3]. This conjecture has been verified for the cases of ∆ ≤ 5. In this paper, we prove that m ≥ 7 4 n when ∆ = 4. It improves the known bound for ∆ = 4 when n > 6.
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