关于边-多重路替换图的1,2,3-猜想和1,2-猜想（英文）

1,2,3-conjecture and 1,2-conjecture for the edge-multiplicity-paths-replacements

设f:E(G)→Z_+是图G的一个边标号,若对G的每个顶点v,c(v)=∑v∈ef(e)定义一个正常的点着色,则称f是邻点可区别的边标号.设g:V(G)∪E(G)→Z_+是图G的一个全标号,若对G的每个顶点v,c(v)=g(v)+∑v∈eg(e)定义一个正常的点着色,则称g是邻点可区别的全标号.对这2个概念的2个猜想分别是1,2,3-猜想(每一个连通图G≠K_2均有用1,2,3进行标号的邻点可区别的边标号)和1,2-猜想(每一个简单图均有用1,2进行标号的邻点可区别的全标号).主要证明了1,2,3-猜想和1,2-猜想对每一个图的边-多重路替换图都是成立的.

Let f：E（G）→Z_＋ be an edge-weighting（labeing）of a graph G.For each v∈V（G）,if c（v）= ∑v∈ef（e） yields a proper coloring of the graph,then fis defined as a neighbour-distinguishing edge labeling of G.Let g：V（G）∪E（G）→Z_＋ be a total-weighting（labeing）of a graph G.For each v∈V（G）,if c（v）=g（v）＋∑v∈eg（e）yields a proper coloring of the graph,then gis defined as a neighbour-distinguishing total labeling of G.For them,there exist two conjectures such as 1,2,3-conjecture（i.e.,every connected graph G≠K_2has a neighbour-distinguishing edge labeling in{1,2,3}）and 1,2-conjecture（i.e.,every simple graph has a neighbour-distinguishing total labeling in{1,2}）.This paper shows that 1,2,3-conjecture and 1,2-conjecture hold for the edge-multiplicity-paths-replacements for any graph.