{"title":"无垂边连通图的判别指标","authors":"W. Imrich, R. Kalinowski, M. Pilsniak, M. Wozniak","doi":"10.26493/1855-3974.1852.4f7","DOIUrl":null,"url":null,"abstract":"We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"The distinguishing index of connected graphs without pendant edges\",\"authors\":\"W. Imrich, R. Kalinowski, M. Pilsniak, M. Wozniak\",\"doi\":\"10.26493/1855-3974.1852.4f7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.\",\"PeriodicalId\":8402,\"journal\":{\"name\":\"Ars Math. Contemp.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ars Math. Contemp.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26493/1855-3974.1852.4f7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.1852.4f7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 7
摘要
我们考虑边缘颜色，不一定是正确的。图G的区分指标D ' (G)是仅靠恒等自同构保持的边着色中颜色的最少个数。已知除三个小环外，对于最大有限次的可数连通图G Δ， D ' (G)≤Δ。我们证明了如果另外的G没有垂边，D ' (G)≤≤≤≤Δ²+ 1。
The distinguishing index of connected graphs without pendant edges
We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.
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