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Neighbor Sum Distinguishing Total Choosability of Cubic Graphs

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Abstract

Let \(G=(V, E)\) be a graph and \({\mathbb {R}}\) be the set of real numbers. For a k-list total assignment L of G that assigns to each member \(z\in V\cup E\) a set \(L_{z}\) of k real numbers, a neighbor sum distinguishing (NSD) total L-coloring of G is a mapping \(\phi :V\cup E \rightarrow {\mathbb {R}}\) such that every member \(z\in V\cup E\) receives a color of \(L_z\), every pair of adjacent or incident members in \(V\cup E\) receive different colors, and \(\sum _{z\in E_{G}(u)\cup \{u\}}\phi (z)\ne \sum _{z\in E_{G}(v)\cup \{v\}}\phi (z)\) for each edge \(uv\in E\), where \(E_{G}(v)\) is the set of edges incident with v in G. In 2015, Pilśniak and Woźniak posed the conjecture that every graph G with maximum degree \(\Delta \) has an NSD total L-coloring with \(L_z=\{1,2,\dots , \Delta +3\}\) for any \(z\in V\cup E\), and confirmed the conjecture for all cubic graphs. In this paper, we extend their result by proving that every cubic graph has an NSD total L-coloring for any 6-list total assignment L.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    Donghan Zhang, You Lu & Shenggui Zhang

  2. College of Mathematics and Computer Applications, Shangluo University, Shangluo, 726000, Shaanxi, China

    Donghan Zhang

  3. Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    You Lu & Shenggui Zhang

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  1. Donghan Zhang
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  2. You Lu
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  3. Shenggui Zhang
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Correspondence to You Lu.

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Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

You Lu’s work was partially supported by the National Natural Science Foundation of China (NO. 11871397) and the Natural Science Basic Research Plan in Shaanxi Province of China (NO. 2020JM-083). Shenggui Zhang’s work was partially supported by the National Natural Science Foundation of China (NO. 11671320 and U1803263) and the Fundamental Research Funds for the Central Universities (NO. 3102019ghjd003).

Appendix

Appendix

\(\%\%\) The NoteBook of Mathematica to compute the coefficients.

\(\%\) INPUT

\(\%\) Lemma 3.1

$$\begin{aligned}&\hbox {P}=(\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_2)(\hbox {x}_1-\hbox {y}_3)(\hbox {y}_1-\hbox {y}_2)(\hbox {y}_1-\hbox {y}_3)(\hbox {y}_2-\hbox {y}_3)(x_1-x_2)\\&\quad (x_1+y_2+y_3-x_2-y_4-y_5)(x_1-x_3) (\hbox {x}_1+\hbox {y}_1+\hbox {y}_3-\hbox {x}_3-\hbox {y}_4-\hbox {y}_6)\\&\quad (\hbox {x}_1-\hbox {x}_4)(\hbox {x}_1+\hbox {y}_1+\hbox {y}_2-\hbox {x}_4-\hbox {y}_5-\hbox {y}_6)\\&\quad (\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_4)(\hbox {x}_2-\hbox {y}_5)(\hbox {y}_1-\hbox {y}_4) (\hbox {y}_1-\hbox {y}_5)(\hbox {y}_4-\hbox {y}_5)(\hbox {x}_2-\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1\\&\quad +\hbox {y}_5-\hbox {x}_3-\hbox {y}_2-\hbox {y}_6)(\hbox {x}_2-\hbox {x}_4)(\hbox {x}_2+\hbox {y}_1+\hbox {y}_4-\hbox {x}_4-\hbox {y}_3-\hbox {y}_6)(\hbox {x}_3-\hbox {y}_2)\\&\quad (\hbox {x}_3-\hbox {y}_4)(\hbox {x}_3-\hbox {y}_6)(\hbox {y}_2-\hbox {y}_4)(\hbox {y}_2-\hbox {y}_6)(\hbox {y}_4-\hbox {y}_6)(\hbox {x}_3-\hbox {x}_4)(\hbox {x}_3\\&\quad +\hbox {y}_2+\hbox {y}_4-\hbox {x}_4-\hbox {y}_3-\hbox {y}_5)(\hbox {x}_4-\hbox {y}_3) (\hbox {x}_4-\hbox {y}_5)(\hbox {x}_4-\hbox {y}_6)(\hbox {y}_3-\hbox {y}_5)(\hbox {y}_3{-}\hbox {y}_6)(\hbox {y}_5{-}\hbox {y}_6);\\ \end{aligned}$$

Cp1=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[P, x\(_1^4\)x\(_2^4\)]] , x\(_3^4\)x\(_4^4\)]], y\(_1^4\)y\(_2^4\)]], y\(_3^4\)y\(_4^4\)y\(_5^4\)]

\(\%\) To calculate the coefficient of x\(_1^4\)x\(_2^4\)x\(_3^4\)x\(_4^4\)y\(_1^4\)y\(_2^4\)y\(_3^4\)y\(_4^4\)y\(_5^4\)

\(\%\) Lemma 3.2 (1)

$$\begin{aligned}&\hbox {P}=(\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_4)(\hbox {x}_1-\hbox {y}_5)(\hbox {y}_1-\hbox {y}_4)(\hbox {y}_1-\hbox {y}_5)(\hbox {y}_4-\hbox {y}_5)\\&\quad (\hbox {x}_1-\hbox {x}_2)(\hbox {x}_1+\hbox {y}_4+\hbox {y}_5-\hbox {x}_2-\hbox {y}_2)(\hbox {x}_1-\hbox {x}_3)\\&\quad (\hbox {x}_1+\hbox {y}_1+\hbox {y}_4-\hbox {x}_3-\hbox {y}_2-\hbox {y}_3)\hbox {x}_2^2\hbox {y}_1\hbox {y}_2(\hbox {x}_2+\hbox {y}_1+\hbox {y}_2)(\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_2)(\hbox {y}_1-\hbox {y}_2)\\&\quad (\hbox {x}_2{-}\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1{-}\hbox {x}_3-\hbox {y}_3{-}\hbox {y}_5)(\hbox {x}_3{-}\hbox {y}_2)(\hbox {x}_3{-}\hbox {y}_3)(\hbox {x}_3{-}\hbox {y}_5)(\hbox {y}_2{-}\hbox {y}_3)(\hbox {y}_2{-}\hbox {y}_5)(\hbox {y}_3{-}\hbox {y}_5)\\&\quad (\hbox {x}_3{-}\hbox {x}_4)(\hbox {x}_3{+}\hbox {y}_2{+}\hbox {y}_5{-}\hbox {x}_4{-}\hbox {y}_4)\hbox {x}_4^2\hbox {y}_3\hbox {y}_4(\hbox {x}_4+\hbox {y}_3+\hbox {y}_4)(\hbox {x}_4{-}\hbox {y}_3)(\hbox {x}_4{-}\hbox {y}_4)(\hbox {y}_3{-}\hbox {y}_4)\\&\quad (\hbox {x}_4-\hbox {x}_1)(\hbox {x}_4+ \hbox {y}_3-\hbox {x}_1-\hbox {y}_1-\hbox {y}_5); \end{aligned}$$

Cp2=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[P, x\(_1^4\)x\(_2^5\)]] , x\(_3^4\)x\(_4^5\)]], y\(_1^4\)y\(_2^4\)]], y\(_3^4\)y\(_4^4\)y\(_5^4\)]

\(\%\) To calculate the coefficient of x\(_1^4\)x\(_2^5\)x\(_3^4\)x\(_4^5\)y\(_1^4\)y\(_2^4\)y\(_3^4\)y\(_4^4\)y\(_5^4\)

\(\%\) Lemma 3.2 (2)

$$\begin{aligned}&\hbox {P}=(\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_4)(\hbox {x}_1-\hbox {y}_5)(\hbox {y}_1-\hbox {y}_4)(\hbox {y}_1-\hbox {y}_5)(\hbox {y}_4-\hbox {y}_5)(\hbox {x}_1-\hbox {x}_2)\\&\quad (\hbox {x}_1+\hbox {y}_4+\hbox {y}_5-\hbox {x}_2-\hbox {y}_2)\hbox {x}_2^2\hbox {y}_1\hbox {y}_2(\hbox {x}_2+\hbox {y}_1+\hbox {y}_2)(\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_2)(\hbox {y}_1-\hbox {y}_2)\\&\quad (\hbox {x}_2-\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1-\hbox {x}_3-\hbox {y}_3-\hbox {y}_6)(\hbox {x}_3-\hbox {y}_2)(\hbox {x}_3-\hbox {y}_3)(\hbox {x}_3-\hbox {y}_6)(\hbox {y}_2-\hbox {y}_3)\\&\quad (\hbox {y}_2-\hbox {y}_6)(\hbox {y}_3-\hbox {y}_6)(\hbox {x}_3-\hbox {x}_4)(\hbox {x}_3+\hbox {y}_2+\hbox {y}_6-\hbox {x}_4-\hbox {y}_4)\hbox {x}_4^2\hbox {y}_3 \hbox {y}_4\\&\quad (\hbox {x}_4+\hbox {y}_3+\hbox {y}_4)(\hbox {x}_4-\hbox {y}_3)(\hbox {x}_4-\hbox {y}_4)(\hbox {y}_3-\hbox {y}_4)\\&\quad (\hbox {x}_4-\hbox {x}_1)(\hbox {x}_4+\hbox {y}_3-\hbox {x}_1-\hbox {y}_1-\hbox {y}_5)\hbox {x}_5^2\hbox {y}_5 \hbox {y}_6(\hbox {x}_5+\hbox {y}_5\\&\quad +\hbox {y}_6)(\hbox {x}_5-\hbox {y}_5)(\hbox {x}_5-\hbox {y}_6)(\hbox {y}_5-\hbox {y}_6)(\hbox {x}_5-\hbox {x}_1)(\hbox {x}_5+\hbox {y}_6-\hbox {x}_1-\hbox {y}_1-\hbox {y}_4)\\&\quad (\hbox {x}_5-\hbox {x}_3) (\hbox {x}_5+\hbox {y}_5-\hbox {x}_3-\hbox {y}_2-\hbox {y}_3); \end{aligned}$$

Cp3=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor [Coefficient[P, x\(_1^4\)x\(_2^5\)]] , x\(_3^5\)x\(_4^5\)]], x\(_5^5\)y\(_1^4\)]], y\(_2^4\)y\(_3^4\)]], y\(_4^4\)y\(_5^3\)y\(_6^5\)]

\(\%\) To calculate the coefficient of x\(_1^4\)x\(_2^5\)x\(_3^5\)x\(_4^5\)x\(_5^5\)y\(_1^4\)y\(_2^4\)y\(_3^4\)y\(_4^4\)y\(_5^3\)y\(_6^5\)

\(\%\) Lemma 3.2 (3) n=3

$$\begin{aligned}&\hbox {P}=\hbox {x}_4^4\hbox {y}_4^2(\hbox {x}_4+\hbox {y}_4)^2(\hbox {x}_4-\hbox {y}_4)\\&\quad (\hbox {x}_4-\hbox {x}_1)(\hbox {x}_4-\hbox {x}_1-\hbox {y}_1-\hbox {y}_3)\\&\quad (\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_3)(\hbox {x}_1-\hbox {y}_4)(\hbox {y}_1-\hbox {y}_3)\\&\quad (\hbox {y}_1-\hbox {y}_4)(\hbox {y}_3-\hbox {y}_4)(\hbox {x}_1-\hbox {x}_2)\\&\quad (\hbox {x}_1+\hbox {y}_3 +\hbox {y}_4-\hbox {x}_2-\hbox {y}_2)\\&\quad \hbox {x}_2^2\hbox {y}_1\hbox {y}_2(\hbox {x}_2+\hbox {y}_1+\hbox {y}_2)\\&\quad (\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_2)(\hbox {y}_1-\hbox {y}_2)\\&\quad (\hbox {x}_2-\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1-\hbox {x}_3-\hbox {y}_3)\\&\quad \hbox {x}_3^2\hbox {y}_2\hbox {y}_3(\hbox {x}_3+\hbox {y}_2\\&\quad +\hbox {y}_3)(\hbox {x}_3-\hbox {y}_2)(\hbox {x}_3-\hbox {y}_3)(\hbox {y}_2-\hbox {y}_3)\\&\quad (\hbox {x}_3-\hbox {x}_1)(\hbox {x}_3+\hbox {y}_2-\hbox {x}_1-\hbox {y}_1-\hbox {y}_4); \end{aligned}$$

Cp4=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[P, x\(_1^5\)y\(_1^5\)]], x\(_2^5\)y\(_2^5\)]], x\(_3^5\)y\(_3^4\)]], x\(_4^5\)y\(_4^5\)]

\(\%\) To calculate the coefficient of x\(_1^5\)y\(_1^5\)x\(_2^5\)y\(_2^5\)x\(_3^5\)y\(_3^4\)x\(_4^5\)y\(_4^5\)

\(\%\) Lemma 3.2 (3) n=4

$$\begin{aligned}&\hbox {P}=\hbox {x}_5^4\hbox {y}_5^2(\hbox {x}_5+\hbox {y}_5)^2(\hbox {x}_5-\hbox {y}_5)(\hbox {x}_5-\hbox {x}_1)(\hbox {x}_5-\hbox {x}_1-\hbox {y}_1-\hbox {y}_4)\\&\quad (\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_4)(\hbox {x}_1-\hbox {y}_5)(\hbox {y}_1-\hbox {y}_4)\\&\quad (\hbox {y}_1-\hbox {y}_5)(\hbox {y}_4-\hbox {y}_5)(\hbox {x}_1-\hbox {x}_2)\\&\quad (\hbox {x}_1+\hbox {y}_4+ \hbox {y}_5-\hbox {x}_2-\hbox {y}_2)\hbox {x}_2^2\hbox {y}_1\hbox {y}_2\\&\quad (\hbox {x}_2+\hbox {y}_1+\hbox {y}_2)(\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_2)\\&\quad (\hbox {y}_1-\hbox {y}_2)(\hbox {x}_2-\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1-\hbox {x}_3-\hbox {y}_3)\\&\quad \hbox {x}_3^2\hbox {y}_2\hbox {y}_3(\hbox {x}_3+\hbox {y}_2\\&\quad +\hbox {y}_3)(\hbox {x}_3-\hbox {y}_2)\\&\quad (\hbox {x}_3-\hbox {y}_3)(\hbox {y}_2-\hbox {y}_3) (\hbox {x}_3-\hbox {x}_4)\\&\quad (\hbox {x}_3+\hbox {y}_2-\hbox {x}_4-\hbox {y}_4)\hbox {x}_4^2\hbox {y}_3\hbox {y}_4(\hbox {x}_4+\hbox {y}_3+\hbox {y}_4)\\&\quad (\hbox {x}_4-\hbox {y}_3)(\hbox {x}_4-\hbox {y}_4)(\hbox {y}_3-\hbox {y}_4)(\hbox {x}_4-\hbox {x}_1)\\&\quad (\hbox {x}_4+\hbox {y}_3-\hbox {x}_1-\hbox {y}_1-\hbox {y}_5); \end{aligned}$$

Cp5=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor [Coefficient[P, x\(_1^5\)y\(_1^5\)]] , x\(_2^5\)y\(_2^5\)]], x\(_3^5\)y\(_3^5\)]], x\(_4^5\)y\(_4^4\)]], x\(_5^5\)y\(_5^5\)]

\(\%\) To calculate the coefficient of x\(_1^5\)y\(_1^5\)x\(_2^5\)y\(_2^5\)x\(_3^5\)y\(_3^5\)x\(_4^5\)y\(_4^4\)x\(_5^5\)y\(_5^5\)

\(\%\) Lemma 3.2 (3) n=5

$$\begin{aligned}&\hbox {P}=\hbox {x}_6^4\hbox {y}_6^2(\hbox {x}_6+\hbox {y}_6)^2(\hbox {x}_6-\hbox {y}_6)(\hbox {x}_6-\hbox {x}_1)(\hbox {x}_6-\hbox {x}_1-\hbox {y}_1-\hbox {y}_5)\\&\quad (\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_5)(\hbox {x}_1-\hbox {y}_6)(\hbox {y}_1-\hbox {y}_5)\\&\quad (\hbox {y}_1-\hbox {y}_6)(\hbox {y}_5-\hbox {y}_6)(\hbox {x}_1-\hbox {x}_2)\\&\quad (\hbox {x}_1+\hbox {y}_5\\&\quad +\hbox {y}_6-\hbox {x}_2-\hbox {y}_2)\hbox {x}_2^2\hbox {y}_1\hbox {y}_2(\hbox {x}_2+\hbox {y}_1+\hbox {y}_2)(\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_2)\\&\quad (\hbox {y}_1-\hbox {y}_2)(\hbox {x}_2-\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1-\hbox {x}_3-\hbox {y}_3)\hbox {x}_3^2\hbox {y}_2\hbox {y}_3(\hbox {x}_3+\hbox {y}_2\\&\quad +\hbox {y}_3)(\hbox {x}_3-\hbox {y}_2)(\hbox {x}_3-\hbox {y}_3)(\hbox {y}_2-\hbox {y}_3)\\&\quad (\hbox {x}_3-\hbox {x}_4)(\hbox {x}_3+\hbox {y}_2-\hbox {x}_4-\hbox {y}_4)\hbox {x}_4^2\hbox {y}_3\hbox {y}_4\\&\quad (\hbox {x}_4+\hbox {y}_3+\hbox {y}_4)(\hbox {x}_4-\hbox {y}_3)(\hbox {x}_4-\hbox {y}_4)(\hbox {y}_3-\hbox {y}_4)\\&\quad (\hbox {x}_4-\hbox {x}_5)(\hbox {x}_4+\hbox {y}_3-\hbox {x}_5-\hbox {y}_5)\hbox {x}_5^2\hbox {y}_4\hbox {y}_5(\hbox {x}_5+\hbox {y}_4+\hbox {y}_5)\\&\quad (\hbox {x}_5-\hbox {y}_4)(\hbox {x}_5-\hbox {y}_5)(\hbox {y}_4-\hbox {y}_5)(\hbox {x}_5-\hbox {x}_1)\\&\quad (\hbox {x}_5+\hbox {y}_4-\hbox {x}_1-\hbox {y}_1-\hbox {y}_6); \end{aligned}$$

Cp6=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor [Coefficient

[Factor[Coefficient[P, x\(_1^5\)y\(_1^5\)]], x\(_2^5\)y\(_2^5\)]], x\(_3^5\)y\(_3^5\)]], x\(_4^5\)y\(_4^5\)]], x\(_5^5\)y\(_5^4\)]], x\(_6^5\)y\(_6^5\)]

\(\%\) To calculate the coefficient of x\(_1^5\)y\(_1^5\)x\(_2^5\)y\(_2^5\)x\(_3^5\)y\(_3^5\)x\(_4^5\)y\(_4^5\)x\(_5^5\)y\(_5^4\)x\(_6^5\)y\(_6^5\)

\(\%\) Lemma 3.2 (3) n=6

$$\begin{aligned}&\hbox {P}=\hbox {x}_7^4\hbox {y}_7^2(\hbox {x}_7+\hbox {y}_7)^2(\hbox {x}_7-\hbox {y}_7)(\hbox {x}_7-\hbox {x}_1)(\hbox {x}_7-\hbox {x}_1-\hbox {y}_1-\hbox {y}_6)\\&\quad (\hbox {x}_1-\hbox {y}_1)(\hbox {x}_1-\hbox {y}_6)(\hbox {x}_1-\hbox {y}_7)(\hbox {y}_1-\hbox {y}_6)\\&\quad (\hbox {y}_1-\hbox {y}_7)(\hbox {y}_6-\hbox {y}_7)(\hbox {x}_1-\hbox {x}_2)\\&\quad (\hbox {x}_1+\hbox {y}_6+\hbox {y}_7-\hbox {x}_2-\hbox {y}_2)\\&\quad \hbox {x}_2^2\hbox {y}_1\hbox {y}_2(\hbox {x}_2+\hbox {y}_1+\hbox {y}_2)\\&\quad (\hbox {x}_2-\hbox {y}_1)(\hbox {x}_2-\hbox {y}_2)\\&\quad (\hbox {y}_1-\hbox {y}_2)(\hbox {x}_2-\hbox {x}_3)(\hbox {x}_2+\hbox {y}_1-\hbox {x}_3-\hbox {y}_3)\\&\quad \hbox {x}_3^2\hbox {y}_2\hbox {y}_3(\hbox {x}_3+\hbox {y}_2\\&\quad +\hbox {y}_3)(\hbox {x}_3-\hbox {y}_2)(\hbox {x}_3-\hbox {y}_3)(\hbox {y}_2-\hbox {y}_3)\\&\quad (\hbox {x}_3-\hbox {x}_4)(\hbox {x}_3+\hbox {y}_2-\hbox {x}_4-\hbox {y}_4)\\&\quad \hbox {x}_4^2\hbox {y}_3\hbox {y}_4(\hbox {x}_4+\hbox {y}_3+\hbox {y}_4)(\hbox {x}_4-\hbox {y}_3)\\&\quad (\hbox {x}_4-\hbox {y}_4)(\hbox {y}_3-\hbox {y}_4)(\hbox {x}_4-\hbox {x}_5)\\&\quad (\hbox {x}_4+\hbox {y}_3-\hbox {x}_5-\hbox {y}_5)\hbox {x}_5^2\hbox {y}_4\hbox {y}_5\\&\quad (\hbox {x}_5+\hbox {y}_4+\hbox {y}_5)(\hbox {x}_5-\hbox {y}_4)(\hbox {x}_5-\hbox {y}_5)\\&\quad (\hbox {y}_4-\hbox {y}_5)(\hbox {x}_5-\hbox {x}_6)(\hbox {x}_5+\hbox {y}_4-\hbox {x}_6-\hbox {y}_6)\\&\quad \hbox {x}_6^2\hbox {y}_5\hbox {y}_6(\hbox {x}_6+\hbox {y}_5\\&\quad +\hbox {y}_6)(\hbox {x}_6-\hbox {y}_5)(\hbox {x}_6-\hbox {y}_6)\\&\quad (\hbox {y}_5-\hbox {y}_6)(\hbox {x}_6-\hbox {x}_1)(\hbox {x}_6+\hbox {y}_5-\hbox {x}_1-\hbox {y}_1-\hbox {y}_7); \end{aligned}$$

Cp7=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor [Coefficient

[Factor[Coefficient[Factor[Coefficient[P, x\(_1^5\)y\(_1^5\)]], x\(_2^5\)y\(_2^5\)]], x\(_3^5\)y\(_3^5\)]], x\(_4^5\)y\(_4^5\)]], x\(_5^5\)y\(_5^5\)]], x\(_6^5\)y\(_6^4\)]], x\(_7^5\)y\(_7^5\)]

\(\%\) To calculate the coefficient of x\(_1^5\)y\(_1^5\)x\(_2^5\)y\(_2^5\)x\(_3^5\)y\(_3^5\)x\(_4^5\)y\(_4^5\)x\(_5^5\)y\(_5^5\)x\(_6^5\)y\(_6^4\)x\(_7^5\)y\(_7^5\)

\(\%\) Lemma 3.2 (4) \(\eta _{J_{0}}\)

$$\begin{aligned}&\hbox {P}=(\hbox {x}_{\mathrm{n}}-\hbox {x}_{1})(\hbox {x}_{\mathrm{n}}+\hbox {y}_{\mathrm{n-1}}+\hbox {y}_{\mathrm{2n}}-\hbox {x}_{1}-\hbox {y}_{1}-\hbox {y}_{n+1})\\&\quad (\hbox {x}_{1}-\hbox {y}_{1})(\hbox {x}_{1}-\hbox {y}_{\mathrm{n}})(\hbox {x}_{1}-\hbox {y}_{n+1})(\hbox {y}_{1}-\hbox {y}_{\mathrm{n}})\\&\quad (\hbox {y}_{1}-\hbox {y}_{\mathrm{n+1}})(\hbox {y}_{\mathrm{n}}-\hbox {y}_{\mathrm{n+1}})\\&\quad (\hbox {x}_{1}-\hbox {x}_{2})(\hbox {x}_{1}+\hbox {y}_{\mathrm{n}}+\hbox {y}_{\mathrm{n+1}}-\hbox {x}_{2}-\hbox {y}_{2}-\hbox {y}_{\mathrm{n+2}})\\&\quad \hbox {x}_{\mathrm{n+1}}^{4}\hbox {y}_{\mathrm{n+1}}^{2}(\hbox {x}_{\mathrm{n+1}}+\hbox {y}_{\mathrm{n+1}})^{2}(\hbox {x}_{\mathrm{n+1}}-\hbox {y}_{\mathrm{n+1}})\\&\quad (\hbox {x}_{\mathrm{n+1}}-\hbox {x}_{1})(\hbox {x}_{\mathrm{n+1}}-\hbox {x}_{1}-\hbox {y}_{1}-\hbox {y}_{n}) (\hbox {x}_{2}\\&\quad -\hbox {y}_{1})(\hbox {x}_{2}-\hbox {y}_{2})(\hbox {x}_{2}-\hbox {y}_{\mathrm{{n+2}}})\\&\quad (\hbox {y}_{1}-\hbox {y}_{2})(\hbox {y}_{1}-\hbox {y}_{\mathrm{{n+2}}})(\hbox {y}_{2}-\hbox {y}_{\mathrm{{n+2}}})\\&\quad (\hbox {x}_{2}-\hbox {x}_{3})(\hbox {x}_{2}+\hbox {y}_{1}+\hbox {y}_{\mathrm{{n+2}}}-\hbox {x}_{3}-\hbox {y}_{3}-\hbox {y}_{\mathrm{{n+3}}})\hbox {x}_{\mathrm{{n+2}}}^{4}\\&\quad \hbox {y}_{\mathrm{{n+2}}}^{2}(\hbox {x}_{\mathrm{{n+2}}} +\hbox {y}_{\mathrm{{n+2}}})^{2}(\hbox {x}_{\mathrm{{n+2}}}-\hbox {y}_{\mathrm{{n+2}}})\\&\quad (\hbox {x}_{\mathrm{{n+2}}}-\hbox {x}_{2})(\hbox {x}_{\mathrm{{n+2}}}-\hbox {x}_{2}-\hbox {y}_{1}-\hbox {y}_{2})\\&\quad (\hbox {x}_{3}-\hbox {y}_{2})(\hbox {x}_{3}-\hbox {y}_{3})\\&\quad (\hbox {x}_{3}-\hbox {y}_{\mathrm{{n+3}}})(\hbox {y}_{2}-\hbox {y}_{3})\\&\quad (\hbox {y}_{2}-\hbox {y}_{\mathrm{{n+3}}})(\hbox {y}_{3}-\hbox {y}_{\mathrm{{n+3}}})\\&\quad (\hbox {x}_{3}-\hbox {x}_{4})(\hbox {x}_{3}+\hbox {y}_{2}+\hbox {y}_{\mathrm{{n+3}}}\\&\quad -\hbox {x}_{4}-\hbox {y}_{4}-\hbox {y}_{\mathrm{{n+4}}})\hbox {x}_{\mathrm{{n+3}}}^{4}\\&\quad \hbox {y}_{\mathrm{{n+3}}}^{2}(\hbox {x}_{\mathrm{{n+3}}}+\hbox {y}_{\mathrm{{n+3}}})^{2}\\&\quad (\hbox {x}_{\mathrm{{n+3}}}-\hbox {y}_{\mathrm{{n+3}}})\\&\quad (\hbox {x}_{\mathrm{{n+3}}}-\hbox {x}_{3})\\&\quad (\hbox {x}_{\mathrm{{n+3}}}-\hbox {x}_{3}-\hbox {y}_{2}-\hbox {y}_{3})\\&\quad (\hbox {x}_{4}-\hbox {y}_{3})(\hbox {x}_{4}-\hbox {y}_{4})\\&\quad (\hbox {x}_{4}-\hbox {y}_{\mathrm{{n+4}}})(\hbox {y}_{3}-\hbox {y}_{4})\\&\quad (\hbox {y}_{3}-\hbox {y}_{\mathrm{{n+4}}})(\hbox {y}_{4}-\hbox {y}_{\mathrm{{n+4}}})\\&\quad (\hbox {x}_{4}-\hbox {x}_{5})\\&\quad (\hbox {x}_{4}+\hbox {y}_{3}+\hbox {y}_{\mathrm{{n+4}}}-\hbox {x}_{5}-\hbox {y}_{5}-\hbox {y}_{\mathrm{{n+5}}})\\&\quad \hbox {x}_{\mathrm{{n+4}}}^{4}\hbox {y}_{\mathrm{{n+4}}}^{2}(\hbox {x}_{\mathrm{{n+4}}}\\&\quad +\hbox {y}_{\mathrm{{n+4}}})^{2}(\hbox {x}_{\mathrm{{n+4}}}-\hbox {y}_{\mathrm{{n+4}}})\\&\quad (\hbox {x}_{\mathrm{{n+4}}}-\hbox {x}_{4})(\hbox {x}_{\mathrm{{n+4}}}-\hbox {x}_{4}-\hbox {y}_{3}-\hbox {y}_{4})\\&\quad (\hbox {x}_{5}-\hbox {y}_{4})(\hbox {x}_{5}-\hbox {y}_{5})\\&\quad (\hbox {x}_{5}-\hbox {y}_{\mathrm{{n+5}}})(\hbox {y}_{4}-\hbox {y}_{5})\\&\quad (\hbox {y}_{4}-\hbox {y}_{\mathrm{{n+5}}})(\hbox {y}_{5}-\hbox {y}_{\mathrm{{n+5}}})\\&\quad (\hbox {x}_{5}-\hbox {x}_{6})(\hbox {x}_{5}+\hbox {y}_{4}+\hbox {y}_{\mathrm{{n+5}}}\\&\quad -\hbox {x}_{6}-\hbox {y}_{6}-\hbox {y}_{\mathrm{{n+6}}})\hbox {x}_{\mathrm{{n+5}}}^{4}\hbox {y}_{\mathrm{{n+5}}}^{2}\\&\quad (\hbox {x}_{\mathrm{{n+5}}}+\hbox {y}_{\mathrm{{n+5}}})^{2}(\hbox {x}_{\mathrm{{n+5}}}-\hbox {y}_{\mathrm{{n+5}}})\\&\quad (\hbox {x}_{\mathrm{{n+5}}}-\hbox {x}_{5})\\&\quad (\hbox {x}_{\mathrm{{n+5}}}-\hbox {x}_{5}-\hbox {y}_{4}-\hbox {y}_{5})\\&\quad (\hbox {x}_{6}-\hbox {y}_{5})(\hbox {x}_{6}-\hbox {y}_{6})\\&\quad (\hbox {x}_{6}-\hbox {y}_{\mathrm{{n+6}}})(\hbox {y}_{5}-\hbox {y}_{6})\\&\quad (\hbox {y}_{5}-\hbox {y}_{\mathrm{{n+6}}})(\hbox {y}_{6}-\hbox {y}_{\mathrm{{n+6}}})\\&\quad (\hbox {x}_{6}-\hbox {x}_{7})(\hbox {x}_{6}+\hbox {y}_{5}+\hbox {y}_{\mathrm{{n+6}}}\\&\quad -\hbox {x}_{7}-\hbox {y}_{7}-\hbox {y}_{n+7})\hbox {x}_{\mathrm{{n+6}}}^{4}\hbox {y}_{\mathrm{{n+6}}}^{2}\\&\quad (\hbox {x}_{\mathrm{{n+6}}}+\hbox {y}_{\mathrm{{n+6}}})^{2}(\hbox {x}_{\mathrm{{n+6}}}-\hbox {y}_{\mathrm{{n+6}}})\\&\quad (\hbox {x}_{\mathrm{{n+6}}}-\hbox {x}_{6})(\hbox {x}_{\mathrm{{n+6}}}-\hbox {x}_{6}-\hbox {y}_{5}-\hbox {y}_{6}) \end{aligned}$$

Cp8=Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor [Coefficient

[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor[Coefficient[Factor [Coefficient

[Factor[Coefficient[P, x\(_{1}^{5}\)x\(_{n+1}^{5}\)]], y\(_{1}^{5}\)y\(_{n+1}^{5}\)]], x\(_{2}^{5}\)x\(_{n+2}^{5}\)]], y\(_{2}^{5}\)y\(_{n+2}^{5}\)]], x\(_{3}^{5}\)x\(_{n+3}^{5}\)]], y\(_{3}^{5}\)y\(_{n+3}^{5}\)]], x\(_{4}^{5}\)x\(_{n+4}^{5}\)]], y\(_{4}^{5}\)y\(_{n+4}^{5}\)]], x\(_{5}^{5}\)x\(_{n+5}^{5}\)]], y\(_{5}^{4}\)y\(_{n+5}^{5}\)]], x\(_{6}^{5}\)x\(_{n+6}^{5}\)y\(_{n+6}^{5}\)]

\(\%\) To calculate the coefficient of x\(_{1}^{5}\)x\(_{n+1}^{5}\)y\(_{1}^{5}\)y\(_{n+1}^{5}\)x\(_{2}^{5}\)x\(_{n+2}^{5}\)y\(_{2}^{5}\)y\(_{n+2}^{5}\) x\(_{3}^{5}\)x\(_{n+3}^{5}\)y\(_{3}^{5}\)y\(_{n+3}^{5}\)x\(_{4}^{5}\)x\(_{n+4}^{5} \)y\(_{4}^{5}\)y\(_{n+4}^{5}\)x\(_{5}^{5}\)x\(_{n+5}^{5}\) y\(_{5}^{4}\)y\(_{n+5}^{5}\) x\(_{6}^{5}\)x\(_{n+6}^{5}\)y\(_{n+6}^{5}\)

\(\%\) Lemma 3.2 (4) \(\eta _{J_{\ell }}\)

$$\begin{aligned}&\hbox {P}=(-3\hbox {x}_{\ell }+3\hbox {y}_{\ell -1}-\hbox {y}_{\ell }-\hbox {y}_{n+\ell })(\hbox {x}_{\ell }-\hbox {y}_{\ell -1})\\&\quad (\hbox {x}_{\ell }-\hbox {y}_{\ell })(\hbox {x}_{\ell }-\hbox {y}_{n+\ell })(\hbox {y}_{\ell -1}-\hbox {y}_{\ell })\\&\quad (\hbox {y}_{\ell -1}-\hbox {y}_{n+\ell })(\hbox {y}_{\ell }-\hbox {y}_{n+\ell })\\&\quad (\hbox {x}_{\ell }-\hbox {x}_{\ell +1})(\hbox {x}_{\ell }+\hbox {y}_{\ell -1}\\&\quad +\hbox {y}_{n+\ell }-\hbox {x}_{\ell +1}-\hbox {y}_{\ell +1}-\hbox {y}_{n+\ell +1})\\&\quad \hbox {x}_{n+\ell }^{4}\hbox {y}_{n+\ell }^{2}(\hbox {x}_{n+\ell }+\hbox {y}_{n+\ell })^{2}(\hbox {x}_{n+\ell }-\hbox {y}_{n+\ell })\\&\quad (\hbox {x}_{n+\ell }-\hbox {x}_{\ell })(\hbox {x}_{n+\ell }-\hbox {x}_{\ell }-\hbox {y}_{\ell -1}-\hbox {y}_{\ell }) \end{aligned}$$

Cp9=Coefficient[Factor[Coefficient[P, x\(_{\ell }^{5}\)x\(_{n+\ell }^{5}\)]], y\(_{\ell -1}^{4}\)y\(_{n+\ell }^{5}\)]

\(\%\) To calculate the coefficient of x\(_{\ell }^{5}\)x\(_{n+\ell }^{5}\)y\(_{\ell -1}^{4}\)y\(_{n+\ell }^{5}\)

\(\%\) OUTPUT

$$\begin{aligned} Cp1= & {} -24\ \ Cp2=4\ \ Cp3=-12\ \ Cp4=2\ \ Cp5=-25\ \ Cp6=18\ \ Cp7=-65\\ Cp8= & {} -46(\hbox {y}_{n-1}+\hbox {y}_{2n})(-3\hbox {x}_7+3\hbox {y}_6-\hbox {y}_7-\hbox {y}_{n+7})\ \ Cp9=3(-3\hbox {x}_{\ell +1}+3\hbox {y}_{\ell }-\hbox {y}_{\ell +1}-\hbox {y}_{n+\ell +1}) \end{aligned}$$

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Zhang, D., Lu, Y. & Zhang, S. Neighbor Sum Distinguishing Total Choosability of Cubic Graphs. Graphs and Combinatorics 36, 1545–1562 (2020). https://doi.org/10.1007/s00373-020-02196-3

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