Neighbor Full Sum Distinguishing Total Coloring of Planar Graphs with Girth at Least 5
A neighbor full sum distinguishing total coloring of a graph <i>G</i> is a proper <i>k</i>-total coloring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics>...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/7/496 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | A neighbor full sum distinguishing total coloring of a graph <i>G</i> is a proper <i>k</i>-total coloring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> such that no two adjacent vertices <i>u</i> and <i>v</i> satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>≠</mo><mi>ω</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><msub><mo>∑</mo><mrow><mi>e</mi><mo>∋</mo><mi>v</mi></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>+</mo><msub><mo>∑</mo><mrow><mi>u</mi><mo>∈</mo><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>u</mi><mo>|</mo><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>}</mo></mrow></semantics></math></inline-formula>. The minimum positive integer <i>k</i> is called the neighbor full sum distinguishing total coloring of <i>G</i>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ftndi</mi><mi>Σ</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for short. In this article, we verify that for a normal planar graph <i>G</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ftndi</mi><mi>Σ</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>Δ</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>5</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>22</mn></mrow></semantics></math></inline-formula>, which extends the result of Yue, et al. by reducing the girth condition from 6 to 5. |
|---|---|
| ISSN: | 2075-1680 |