关于小平衡图、强平衡图和全图
On small balanceable, strongly-balanceable and omnitonal graphs
摘要：
在Ramsey理论中，我们给出了一个图$G$，我们需要找到最小的$n_0$，使得对于任何$n\geq n_0$，$K_n$的边的任何红色/蓝色都会给出一个子图$G$，其所有边都是蓝色或都是红色。这里我们要求，对于$K_n$边缘的任何红色/蓝色着色，必须有一个$G$的副本，使得它的边缘被相等地划分为红色或蓝色（或者在$G$具有奇数个边缘的情况下，颜色类的大小相差一个）。这引入了可平衡图的概念和$G$的平衡数，如果存在的话，该平衡数是最小整数bal $(n, G)$，使得对于具有多于bal $(n, G)$的任一颜色的边缘的$E(K_n)$的任何红/蓝着色，$K_n$将包含如上所述的$G$的平衡着色副本。该参数由Caro、Hansberg和Montejano在\cite{2018arXivCHM}中引入。在那里，作者还介绍了强平衡数sbal $(n,G)$和更一般的omnitonal数的$(n, G)$，它要求$G$的副本包含一个完整的分布的红色和蓝色边缘的数量在$E(G)$。本文将对至多四条边上的所有图$G$，分类bal $(n, G)$，sbal $(n, G)$和ot $(n,G)$.我们将使用Caro等人的一些关键结果，我们在这里完全重现，以及我们在这里证明的一些新结果。例如，我们将证明具有相同边数的两个二部图的并总是平衡的。
英文摘要：
In Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red. Here we shall be requiring that, for any red/blue colouring of the edges of $K_n$, there must be a copy of $G$ such that its edges are partitioned equally as red or blue (or the sizes of the colour classes differs by one in the case when $G$ has an odd number of edges). This introduces the notion of balanceable graphs and the balance number of $G$ which, if it exists, is the minimum integer bal$(n, G)$ such that, for any red/blue colouring of $E(K_n)$ with more than bal$(n, G)$ edges of either colour, $K_n$ will contain a balanced coloured copy of $G$ as described above. This parameter was introduced by Caro, Hansberg and Montejano in \cite{2018arXivCHM}. There, the authors also introduce the strong balance number sbal$(n,G)$ and the more general omnitonal number ot$(n, G)$ which requires copies of $G$ containing a complete distribution of the number of red and blue edges over $E(G)$. In this paper we shall catalogue bal$(n, G)$, sbal$(n, G)$ and ot$(n,G)$ for all graphs $G$ on at most four edges. We shall be using some of the key results of Caro et al, which we here reproduce in full, as well as some new results which we prove here. For example, we shall prove that the union of two bipartite graphs with the same number of edges is always balanceable.
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