路、圈的广义Mycielski图的邻点可区别Ⅰ-全色数  

Adjacent Vertex-Distinguishing Ⅰ-total Colorings of Mycielski's Graphs of Path and Cycle

作　　者：杨随义[1] YANG Sui-yi(Department of Mathematics,Tianshui Normal University,Tianshui 741001,China)

机构地区：[1]天水师范学院数学与统计学院,甘肃天水741000

出　　处：《数学的实践与认识》2023年第5期142-152,共11页Mathematics in Practice and Theory

基　　金：甘肃省自然科学基金(20JR5RA498,21JR11RA065)。

摘　　要：图G的邻点可区别Ⅰ-全染色是一个满足相邻顶点色集合不同的Ⅰ-全染色,其中任意一点的色集合包含该顶点及其关联边所染的颜色.所需颜色的最小数称为邻点可区别Ⅰ-全色数,记作χ_(at)^(i)(G).研究了路和圈的广义Mycielski图的邻点可区别Ⅰ-全色数:对于阶数n≥2的路P_(n),当n=2,3,4时,有χ_(at)^(i)(M(P_(n)))=n+1;否则,χ_(at)^(i)(M(Pn))=n.对于阶数n≥3的圈Cn,当n=3,4时,有χ_(at)^(i)(M(Cn))=5;否则,χ_(at)^(i)(M(Cn))=n.An adjacent vertex-distinguishing Ⅰ-total coloring of a graph is an Ⅰ-total coloring such that no two adjacent vertices have the same color-set,where the color-set of each vertex consists of all colors assigned on the vertex and its incident edges.The minimum number required for a adjacent vertex-distinguishing Ⅰ-total coloring of a graph G is called the adjacent vertex-distinguishing Ⅰ-total chromatic number,which is denoted by χ_(at)^(i)(G).In this paper,the adjacent vertex-distinguishing Ⅰ-total colorings of generalized Mycielski's graph such as path and cycle are studied.Exactly,for a path Pn(n≥2),we prove that χ_(at)^(i)(M(Pn))=n+1 if n=2,3,4;and χ_(at)^(i)(M(Pn)=n)otherwise;For a cycle Cn(n≥3),we prove that χ_(at)^(i)(M(Cn))=5 if n=3,4;and χ_(at)^(i)(M(Pn))=n otherwise.

关 键 词：广义MYCIELSKI图 邻点可区别Ⅰ-全染色 邻点可区别Ⅰ-全色数 

分 类 号：O157.5[理学—数学]