发表机构
Linköping University; Yerevan State University（林雪平大学; 埃里温国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出图的局部亏格和弱局部亏格，度量图偏离区间可染色的程度，证明其与亏格差异可任意大，并给出构造性结果与开放问题。
AI 中文摘要
图的区间边染色是指用整数对边进行正常染色，使得与任一顶点关联的边的颜色构成一个整数区间。并非所有图都是区间可染色的；一个简单的反例是$K_3$。图$G$的（区间染色）亏格是使得在$G$上添加悬挂边后得到的图具有区间边染色所需的最少悬挂边数。本文引入并研究了衡量图偏离区间可染色程度的更多度量。图$G$的局部亏格是指在$G$的每个顶点处添加所需的最少悬挂边数，以便获得具有区间边染色的图；我们可以将这些添加边的颜色视为顶点处“局部缺失”的颜色。我们还研究了该概念的较弱版本，即弱局部亏格，非正式地讲，它是在$G$的某个正常边染色中，使得该大小最小化的顶点处“局部缺失”的最大连续整数集合的大小。我们比较了弱局部亏格、局部亏格和亏格，并表明在这两种情况下，差异可以任意大。此外，我们给出了具体的图例，其弱局部亏格（从而局部亏格）随顶点数和最大度数的增加而增长。我们还证明了关于弱局部亏格较小的图的一些构造性结果。特别地，所有完全多部图的弱局部亏格至多为$2$，许多完全多部图的弱局部亏格至多为$1$。此外，最大度数至多为$6$的二部图和最大度数至多为$8$的欧拉二部图的弱局部亏格都至多为$1$。我们最后指出了若干待研究的开放问题。
英文摘要
An interval edge coloring of a graph is a proper edge coloring by integers such that the colors on the edges incident with any vertex form an interval of integers. Not all graphs are interval colorable; a simple counterexample is $K_3$. The (interval coloring) deficiency of a graph $G$ is the minimum number of pendant edges whose addition to $G$ yields a graph with an interval edge coloring. In this paper, we introduce and study further measures of how far from being interval colorable a graph is. The local deficiency of a graph $G$ is the smallest number of pendant edges that needs to be added at every vertex of $G$ in order to obtain a graph with an interval edge coloring; we can think of the colors of these added edges as ''locally missing'' at a vertex. We also study a weaker version of this notion, the weak local deficiency, which informally is the size of a largest set of consecutive integers ''locally missing'' at a vertex in a proper edge coloring of $G$ minimizing this size. We compare weak local deficiency, local deficiency, and deficiency, and show that the difference can be arbitrarily large in both cases. Moreover, we give concrete examples of graphs whose weak local deficiency (and thus local deficiency) grows with the number of vertices as well as with the maximum degree. We also prove some constructive results on graphs with small weak local deficiency. In particular, all complete multipartite graphs have weak local deficiency at most $2$, and many complete multipartite graphs have weak local deficiency at most $1$. Moreover, bipartite graphs with maximum degree at most $6$, and Eulerian bipartite graphs with maximum degree at most $8$ both have weak local deficiency at most $1$. We conclude the paper by pointing to several open questions for further research.
