Asymptotic Expansions for Sub-Critical Lagrangean Forms

Asymptotic expansions for the Taylor coefficients of the Lagrangean form phi(z)=zf(phi(z)) are examined with a focus on the calculations of the asymptotic coefficients. The expansions are simple and useful, and we discuss their use in some enumerating sequences in trees, lattice paths and planar maps

On the precise value of the strong chromatic-index of a planar graph with a large girth

一個圖 $G$ 的 $k$-強邊著色指的是使得距離為二以內的邊都塗不同顏色的 $k$-邊著色；強邊著色數 $chi''_s(G)$ 則標明參數 $k$ 的最小可能。此概念最初是為了解決平地上設置廣播網路的問題，由 Fouquet 與 Jolivet 提出。對於任意圖 $G$，參數 $sigma(G)=max_{xyin E(G)}{deg(x)+deg(y)-1}$是強邊著色數的一個下界；且若 $G$ 是樹，則強邊著色數會到達此下界。另一方面，對於最大度數為 $Delta$ 的平面圖$G$，經由四色定理可以證得 $chi''_s(G)leq 4Delta+4$。更進一步，在各種腰圍與最大度數的條件下，平面圖的強邊著色數之上界分別有$4Delta$, $3Delta+5$, $3Delta+1$, $3Delta$ 和 $2Delta-1$ 等等優化。本篇論文說明當平面圖 $G$ 的腰圍夠大，且$sigma(G)geqDelta(G)+2$ 時，參數 $sigma(G)$ 就會恰好是此圖的強邊著色數。本結果反映出大腰圍的平面圖局部上有看似樹的結構。A {em strong $k$-edge-coloring} of a graph $G$ is a mapping from the edge set $E(G)$ to ${1,2,ldots,k}$ such that every pair of distinct edges at distance at most two receive different colors. The {it strong chromatic index} $chi''_s(G)$ of a graph $G$ is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. The concept of strong edge-coloring was introduced by Fouquet and Jolivet to model the channel assignment in some radio networks. Denote the parameter $sigma(G)=max_{xyin E(G)}{deg(x)+deg(y)-1}$. It is easy to see that $sigma(G) le chi''_s(G)$ for any graph $G$, and the equality holds when $G$ is a tree. For a planar graph $G$ of maximum degree $Delta$, it was proved that $chi''_s(G) le 4 Delta +4$ by using the Four Color Theorem. The upper bound was then reduced to $4Delta$, $3Delta+5$, $3Delta+1$, $3Delta$, $2Delta-1$ under different conditions for $Delta$ and the girth. In this paper, we prove that if the girth of a planar graph $G$ is large enough and $sigma(G)geq Delta(G)+2$, then the strong chromatic index of $G$ is precisely $sigma(G)$. This result reflects the intuition that a planar graph with a large girth locally looks like a tree