关于Gupta的余密度猜想
On Gupta's Co-density Conjecture
摘要：
设$G=(V,E)$是一个重图。$G$的{ \em覆盖索引} $ξ(G)$是最大整数$k$，对于该最大整数，存在具有$k$颜色的$E$的着色，使得$G$的每个顶点与每个颜色的至少一个边关联。令$δ(G)$是$G$的最小度，令$Φ(G)$是$G$的{ \em余密度}，由\[Φ(G)=\min \Big\{\frac{2|E^+(U)|}{|U|+1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},\]定义，其中$E^+(U)$是$G$的所有边的集合，其中至少一端在$U$中。很容易看出，$ξ(G) \le \min\{δ(G), \lfloor Φ(G) \rfloor\}$。1978年Gupta提出了如下的余密度猜想：每个重图$G$满足$ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$，这是关于重图边着色的Goldberg-Seymour猜想的对偶版本。本文证明了当$Φ(G)$不是整数时，$ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$是整数;否则，$ξ(G)\ge \min\{δ(G)-2, \, \lfloor Φ(G) \rfloor-1\}$是整数.我们还表明，这个共密度猜想意味着另一个猜想，由古普塔在1967年作出的覆盖指数。
英文摘要：
Let $G=(V,E)$ be a multigraph. The {\em cover index} $ξ(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. Let $δ(G)$ be the minimum degree of $G$ and let $Φ(G)$ be the {\em co-density} of $G$, defined by \[Φ(G)=\min \Big\{\frac{2|E^+(U)|}{|U|+1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},\] where $E^+(U)$ is the set of all edges of $G$ with at least one end in $U$. It is easy to see that $ξ(G) \le \min\{δ(G), \lfloor Φ(G) \rfloor\}$. In 1978 Gupta proposed the following co-density conjecture: Every multigraph $G$ satisfies $ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$, which is the dual version of the Goldberg-Seymour conjecture on edge-colorings of multigraphs. In this note we prove that $ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$ if $Φ(G)$ is not integral and $ξ(G)\ge \min\{δ(G)-2, \, \lfloor Φ(G) \rfloor-1\}$ otherwise. We also show that this co-density conjecture implies another conjecture concerning cover index made by Gupta in 1967.
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