We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge contraction, we obtain new upper bounds on the chromatic number in terms of the number of induced cycles and the Hadwiger number in terms of the number of induced non-separating cycles. As an application, we show that every 3-connected graph with average degree at least 2k for some \(k\ge 2\) have at least \((k-1)|V|+Ck^{3}\log ^{3/2}k\) induced non-separating cycles for some explicit constant \(C>0\). This improves the previous best known lower bound \((k-1)|V|+1\), which follows from Tutte’s cycle space theorem. We also give a short proof of this theorem of Tutte.

我们研究了由欧拉特征获得的两个参数，方法是用诱导和诱导非分离循环替换面数。通过在一定的同态和边缘收缩下建立此类参数的单调性，我们在色数上以诱导循环数为单位，在Hadwiger数上以诱导非分离循环数为单位获得新的上界。作为一个应用程序，我们证明对于某些\（k \ ge 2 \），平均度至少为2 k的每个3连通图至少具有\（（k-1）| V | + Ck ^ {3} \ log ^ {3/2} k \）诱导了一些明确常数\（C> 0 \）的非分离循环。这改善了先前最著名的下界\（（k-1）| V | +1 \），这是根据Tutte的循环空间定理得出的。我们还简要证明了Tutte的这一定理。