限制最大平均度下图的邻点全和可区别全染色
Neighbor full sum distinguishing total coloring of graphs with bounded maximum average degree
- 中山大学学报(自然科学版)(中英文) 2026年65卷第1期 页码：135-143
- 作者机构：
兰州交通大学应用数学研究所， 甘肃 兰州 730070
- 作者简介：
李志军（1998年生），男；研究方向：图染色；E-mail：lzj12282023@163.com
文飞（1984年生），男；研究方向：图染色、图谱理论；E-mail：wenfei@lzjtu.edu.cn
- 基金信息：国家自然科学基金(12261055);甘肃省自然科学基金重点项目(24JRRA222)
DOI：10.13471/j.cnki.acta.snus.ZR20240354
中图分类号： O157.5收稿：2025-12-17，
修回：2025-09-23，
录用：2025-09-23，
网络首发：2025-11-06，
纸质出版：2026-01-25
- 稿件说明：
移动端阅览
李志军,文飞,岳忠政.限制最大平均度下图的邻点全和可区别全染色[J].中山大学学报(自然科学版)(中英文),2026,65(01):135-143.
LI Zhijun,WEN Fei,YUE Zhongzheng.Neighbor full sum distinguishing total coloring of graphs with bounded maximum average degree[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2026,65(01):135-143.
李志军,文飞,岳忠政.限制最大平均度下图的邻点全和可区别全染色[J].中山大学学报(自然科学版)(中英文),2026,65(01):135-143. DOI： 10.13471/j.cnki.acta.snus.ZR20240354.
LI Zhijun,WEN Fei,YUE Zhongzheng.Neighbor full sum distinguishing total coloring of graphs with bounded maximum average degree[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2026,65(01):135-143. DOI： 10.13471/j.cnki.acta.snus.ZR20240354.
摘要
图
<math id="M1"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519627&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519639&type=
2.03200006
2.28600001
的邻点全和可区别全染色是一个满足任意两个相邻顶点的全和不同的正常全染色，其中顶点的全和是该点及其关联元素所染的颜色之和. 将所用的最少颜色数称为
<math id="M2"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519627&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519639&type=
2.03200006
2.28600001
的邻点全和可区别全色数，简记为
<math id="M3"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519628&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519640&type=
11.76866722
3.38666677
. 设
<math id="M4"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519602&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519642&type=
10.24466610
3.21733332
和
<math id="M5"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519618&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519644&type=
6.94266701
3.21733332
分别表示图
<math id="M6"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519627&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519639&type=
2.03200006
2.28600001
的最大平均度和最大度，运用组合零点定理与权转移法得到： （i）若
<math id="M7"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">3</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519605&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519604&type=
12.44600010
3.21733332
且
<math id="M8"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">12</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519635&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519634&type=
17.86466599
6.26533365
，则
<math id="M9"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">6</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519668&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519622&type=
17.27199936
3.38666677
； （ii）若
<math id="M10"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">4</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519698&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519649&type=
12.44600010
3.21733332
且
<math id="M11"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">11</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519652&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519651&type=
17.86466599
6.26533365
， 则
<math id="M12"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">8</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519714&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519673&type=
17.27199936
3.38666677
； （iii）若
<math id="M13"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">5</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519716&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519703&type=
12.44600010
3.21733332
且
<math id="M14"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">7</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519705&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519704&type=
16.34066582
6.26533365
， 则
<math id="M15"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><mn mathvariant="normal">9</mn><mo>
</mo><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519708&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519693&type=
39.87799835
3.38666677
. 从而推知，当图
<math id="M16"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519627&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519639&type=
2.03200006
2.28600001
满足
<math id="M17"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">7</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519696&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519695&type=
12.44600010
3.21733332
且
<math id="M18"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">7</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519712&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519730&type=
16.34066582
6.26533365
时，邻点全和可区别全染色猜想（崔福祥等， 2023）是成立的.
Abstract
A neighbor full sum distinguishing total coloring of a graph
<math id="M19"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519733&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519732&type=
2.37066650
2.62466669
is a proper total-coloring such that no two adjacent vertices meet the same full sum， where the full sum of a vertex consists of all colors assigned on the vertex and its incident elements， the minimum number of the colors required is called the neighbor full sum distinguishing total chromatic number， denoted by
<math id="M20"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519773&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519746&type=
13.63133430
3.89466691
. Let
<math id="M21"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519775&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519774&type=
11.93799973
3.72533321
and
<math id="M22"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519777&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519776&type=
8.12800026
3.72533321
denote the maximum average degree and the maximum degree of graph
<math id="M23"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519733&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519732&type=
2.37066650
2.62466669
， respectively. In this paper， we prove the following results. （i） If
<math id="M24"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519733&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519732&type=
2.37066650
2.62466669
is a graph with
<math id="M25"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">3</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519754&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519778&type=
14.47799969
3.72533321
and
<math id="M26"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">12</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519770&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519769&type=
20.91266632
7.36600018
， then
<math id="M27"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">6</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519818&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519772&type=
20.06599998
3.89466691
. （ii） If
<math id="M28"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519733&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519732&type=
2.37066650
2.62466669
is a graph with
<math id="M29"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">4</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519820&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519819&type=
14.47799969
3.72533321
and
<math id="M30"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">11</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519822&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519821&type=
20.91266632
7.36600018
， then
<math id="M31"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≤</mo><mn mathvariant="normal">8</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519798&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519823&type=
20.06599998
3.89466691
. （iii） If
<math id="M32"><mi>G</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519733&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519732&type=
2.37066650
2.62466669
is a graph with
<math id="M33"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">5</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519837&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519799&type=
14.56266594
3.72533321
and
<math id="M34"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">7</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519828&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519827&type=
19.04999924
7.36600018
， then
<math id="M35"><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≤</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><mn mathvariant="normal">9</mn><mo>
</mo><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519830&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519829&type=
46.48199844
3.89466691
. By the way， it further follows that the conjecture（Cui et al.， 2023） of neighbor full sum distinguishing total coloring holds on graphs with
<math id="M36"><mi>Δ</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">7</mn></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519832&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519843&type=
14.56266594
3.72533321
and
<math id="M37"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced><mo><</mo><mfrac><mrow><mn mathvariant="normal">7</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519868&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=98519845&type=
19.04999924
7.36600018
.
关键词
Keywords
references
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