关于$ω$和$χ$的多部分分解凸组合的界
Some bounds on convex combinations of $ω$ and $χ$ for decompositions into many parts
摘要：
完全图$K_n$的\emph{$k$--decomposition}是$K_n$分解成$k$生成子图$G_1,...,G_k$。对于图参数$p$，令$p(k;K_n)$表示$K_n$的所有$k$ -分解上的$\displaystyle \sum_{j=1}^{k} p(G_j)$的最大值。已知$χ(k;K_n) = omega(k;K_n)$对$k \leq 3$成立，并证明该等式对所有$k$成立。为了解决这个问题，我们研究了$ω$和$χ$的凸组合，即$0 \leq r \leq 1$的图形参数$A_r(G) = (1-r) ω(G) + r χ(G)$。证明了$A_r(k;K_n) \leq n + {k \choose 2}$为小$r$。此外，我们证明了Kostochka等人的一个定理的一些推广。\cite{kostochka}。
英文摘要：
A \emph{$k$--decomposition} of the complete graph $K_n$ is a decomposition of $K_n$ into $k$ spanning subgraphs $G_1,...,G_k$. For a graph parameter $p$, let $p(k;K_n)$ denote the maximum of $\displaystyle \sum_{j=1}^{k} p(G_j)$ over all $k$--decompositions of $K_n$. It is known that $χ(k;K_n) = omega(k;K_n)$ for $k \leq 3$ and conjectured that this equality holds for all $k$. In an attempt to get a handle on this, we study convex combinations of $ω$ and $χ$; namely, the graph parameters $A_r(G) = (1-r) ω(G) + r χ(G)$ for $0 \leq r \leq 1$. It is proven that $A_r(k;K_n) \leq n + {k \choose 2}$ for small $r$. In addition, we prove some generalizations of a theorem of Kostochka, et al. \cite{kostochka}.
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