A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\ge 2^{1-e(H)}$ holds for every graphon $W:{[0,1]}^2\to [0,1]$, where $t_H(.)$ denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989).

We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, $t_H(W)+t_H(1-W)\ge t_{K_2}{(W)}^{e(H)}+t_{K_2}{(1-W)}^{e(H)}$ whenever H is a path or a cycle and $W:{[0,1]}^2\to \mathbb{R}$ is a bounded symmetric measurable function.

This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons W and odd cycles H. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.

如果完整图的 2 边着色中H的单色副本的数量，则图H是常见的$K_n$通过随机着色渐近最小化，或者等效地，$t_H（瓦）+t_H（1-瓦）\ge 2^{1-e（H）}$对于每个图子都成立$瓦：{[0,1]}^2\to [0,1]$， 在哪里$t_H（。）$表示图H的同态密度。路径和循环的共同性是极值图论最早的基石之一，这要归功于 Mulholland 和 Smith (1959)、Goodman (1959) 和 Sidorenko (1989)。

我们证明了图同态不等式，它扩展了路径和循环的共性。即，$t_H（瓦）+t_H（1-瓦）\ge t_{K_2}{（瓦）}^{e（H）}+t_{K_2}{（1-瓦）}^{e（H）}$每当H是路径或循环并且$瓦：{[0,1]}^2\to \mathbb{右}$是有界对称可测函数。

这回答了 Sidorenko 1989 年提出的问题，他证明了偶数长度路径的稍弱结果来证明奇数循环的共性。此外，它还以强形式解决了 Behague、Morrison 和 Noel 最近的猜想，他们询问不等式是否适用于图子W和奇数周期H 。我们的证明使用完全齐次对称函数的 Schur 凸性，这可能是独立的兴趣。