In 1959, Grötzsch [5] famously proved that every planar graph of girth at least 4 is 3-colourable (or equivalently, admits a homomorphism to C₃). A natural generalization of this is the following conjecture: for every positive integer t, every planar graph of girth at least 4t admits a homomorphism to C_2t+1. This is in fact the planar dual of a well-known conjecture of Jaeger [7] which states that every 4t-edge-connected graph admits a modulo (2t + 1)-orientation. Though Jaeger’s original conjecture was disproved in [6], Lovász et al. [10] showed that every 6t-edge connected graph admits a modolo (2t + 1)-flow. The latter result implies that every planar graph of girth at least 6t admits a homomorphism to C_2t+1. We improve upon this in the t = 3 case, by showing that every planar graph of girth at least 16 admits a homomorphism to C₇. We obtain this through a more general result regarding the density of C₇-critical graphs: if G is a C₇-critical graph with G ∉ {C₃, C₅}, then \(e(G) \ge {{17v(G) - 2} \over {15}}\).

1959 年，Grötzsch [5] 著名地证明了每个周长至少为 4 的平面图是 3 可着色的（或等效地，承认C ₃的同态）。对此的自然推广是以下猜想：对于每个正整数t ，每个周长至少为 4 t的平面图都承认C _{2 t +1}的同态。这实际上是 Jaeger [7] 的一个著名猜想的平面对偶，该猜想指出每个 4 t边连通图都允许模 (2 t + 1) 方向。尽管 Jaeger 的原始猜想在 [6] 中被推翻，但 Lovász 等人。[10] 表明，每 6 个t边连通图承认一个 modolo (2t + 1)-流量。后一个结果意味着每个周长至少为 6 t的平面图都承认 C _{2 t +1}的同态。我们在t = 3 的情况下对此进行了改进，通过显示每个周长至少为 16 的平面图承认C ₇的同态。我们通过关于C ₇ -临界图的密度的更一般的结果来获得这一点：如果 G 是具有G ∉ { C ₃ , C ₅ }的C ₇ -临界图，则\(e(G) \ge {{17v (G) - 2} \over {15}}\)。