On the Alon–Tarsi number of semi-strong product of graphs

Abstract

The Alon–Tarsi number was defined by Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). The Alon–Tarsi number AT(G) of a graph G is the smallest integer k such that G has an orientation D with maximum outdegree $k-1$ and the number of even circulation is not equal to that of odd circulations in D. It is known that $\chi (G)\le \chi _l(G)\le AT(G)$ for any graph G, where $\chi (G)$ and $\chi _l(G)$ are the chromatic number and the list chromatic number of G. Denote by $H_1 \square H_2$ and $H_1\bowtie H_2$ the Cartesian product and the semi-strong product of two graphs $H_1$ and $H_2$, respectively. Kaul and Mudrock (Electron J Combin 26(1):P1.3, 2019) proved that $AT(C_{2k+1}\square P_n)=3$. Li, Shao, Petrov and Gordeev (Eur J Combin 103697, 2023) proved that $AT(C_n\square C_{2k})=3$ and $AT(C_{2m+1}\square C_{2n+1})=4$. Petrov and Gordeev (Mosc. J. Comb. Number Theory 10(4):271–279, 2022) proved that $AT(K_n\square C_{2k})=n$. Note that the semi-strong product is noncommutative. In this paper, we determine $AT(P_m \bowtie P_n)$, $AT(C_m \bowtie C_{2n})$, $AT(C_m \bowtie P_n)$ and $AT(P_m \bowtie C_{n})$. We also prove that $5\le AT(C_m \bowtie C_{2n+1})\le 6$.

Appendix

Procedure 1

$$\begin{aligned}{} & {} >>syms\ \ x1\ \ x2\ \ x3\ \ x4\ \ x5\ \ x6\ \ x9\ \ x10\\{} & {} >> f=(x1-x3)*(x1-x4)*(x2-x3)*(x2-x4)*(x3-x5)\\{} & {} \quad *(x3-x6)*(x4-x5)*(x4-x6);\\{} & {} >> f1=diff(diff(f,x3,2),x4,2)/factorial(2)\wedge 2; x3=0;x4=0;f1\\{} & {} \quad =factor(subs(f1)) \end{aligned}$$

$f1 =(x2*x6+x2*x5+x1*x2+x1*x5+x6*x5+x1*x6)\wedge 2$

$$\begin{aligned}{} & {} >> f2=(x1*x2+x1*x5+x1*x6+x2*x5+x2*x6+x5*x6)\wedge 2\\{} & {} \quad *(x5*x6+x5*x9+x5*x10+x6*x9+x6*x10+x9*x10)\wedge 2\\{} & {} \quad *(x1-x10)*(x1-x9)*(x2-x9)*(x2-x10);\\{} & {} >> f3=diff(diff(diff(diff(f2,x1,2),x2,2),x9,2),x10,2)/factorial(2)\wedge 4;\\{} & {} >> x1=0;x2=0;x9=0;x10=0;f3=expand(subs(f3)) \end{aligned}$$

$f3 =56*x6*x5\wedge 3+90*x6\wedge 2*x5\wedge 2+56*x6\wedge 3*x5+16*x5\wedge 4+16*x6\wedge 4$

$>> diff(diff(f3,x5,2),x6,2)/factorial(2)\wedge 2$

$ans =90$

Procedure 2

$$\begin{aligned}{} & {} >> syms\ \ x1\ \ x2\ \ x3\ \ x4\ \ x5\ \ x6\ \ x7\ \ x8\ \ x9\ \ x10\ \ x11\ \ x12\ \ x13\ \ x14\\{} & {} >>f=(x1-x14)*(x1-x13)*(x2-x14)*(x2-x13)*\\{} & {} \quad (x1*x2+x1*x5+x1*x6+x2*x5+x2*x6+x5*x6)\wedge 2*\\{} & {} \quad (x9*x10+x9*x13+x9*x14 +x10*x13+x10*x14+x13*x14)\wedge 2;\\{} & {} >>f1= diff(diff(diff(diff(f,x1,2),x2,2),x13,2),x14,2)/factorial(2)\wedge 4;\\{} & {} >> x1=0;x2=0;x13=0;x14=0; f1=expand(subs(f1)) \end{aligned}$$

$f1 =-16*x6\wedge 2*x5*x10-16*x6*x5\wedge 2*x10-16*x6\wedge 2*x5*x9+44*x6*x5*x10\wedge 2+44*x6*x5*x9\wedge 2 +44*x9*x10*x6\wedge 2+44*x9*x10*x5\wedge 2-16*x9*x10\wedge 2*x6-16*x9\wedge 2*x10*x6-16*x9*x10\wedge 2*x5 -16*x9\wedge 2*x10*x5-16*x6*x5\wedge 2*x9+16*x5\wedge 2*x9\wedge 2+16*x5\wedge 2*x10\wedge 2 +16*x6\wedge 2*x9\wedge 2+16*x6\wedge 2*x10\wedge 2+x9\wedge 2*x10\wedge 2+120*x6*x5*x9*x10+x6\wedge 2*x5\wedge 2$

$$\begin{aligned}{} & {} >> f2=(x5*x6+x5*x9+x5*x10+x6*x9+x6*x10+x9*x10)\wedge 2*f1;\\{} & {} >> diff(diff(diff(diff(f2,x5,2),x6,2),x9,2),x10,2)/factorial(2)\wedge 4 \end{aligned}$$

$ans =882$

Procedure 3

$$\begin{aligned}{} & {} >>syms\ \ x5\ \ x6\ \ x9\ \ x10\ \ x13\ \ x14\\{} & {} >> f=(x5*x6+x5*x9+x5*x10+x6*x9+x6*x10+x9*x10)\wedge 2*\\{} & {} \quad (x9*x10+x9*x13+x9*x14+x10*x13+x10*x14+x13*x14)\wedge 2;\\{} & {} >> f1=diff(diff(f,x9,2),x10,2)/factorial(2)\wedge 2; x9=0;x10=0; f1\\{} & {} \quad =expand(subs(f1)) \end{aligned}$$

$f1=8*x13\wedge 2*x14*x5+44*x6*x5*x13*x14+x6\wedge 2*x5\wedge 2+x13\wedge 2*x14\wedge 2+8*x13*x14\wedge 2*x5 +8*x13\wedge 2*x14*x6+8*x13*x14\wedge 2*x6+16*x5\wedge 2*x13*x14+16*x6*x5*x13\wedge 2+16*x6*x5*x14\wedge 2 +16*x6\wedge 2*x13*x14+8*x6*x5\wedge 2*x13+8*x6\wedge 2*x5*x13+8*x6*x5\wedge 2*x14+8*x6\wedge 2*x5*x14 +6*x5\wedge 2*x13\wedge 2+6*x5\wedge 2*x14\wedge 2+6*x6\wedge 2*x13\wedge 2+6*x6\wedge 2*x14\wedge 2$

Without loss of generality, let $t=5$.

$$\begin{aligned}{} & {} >> syms\ \ x5\ \ x6\ \ x13\ \ x14\ \ x15\ \ x16\ \ x17\ \ x18\ \ A\ \ B\ \ C\ \ D\ \ E\ \ F\\{} & {} >>f1=A*(x5*x6\wedge 2*x13+x5\wedge 2*\\{} & {} \quad x6*x13+x5*x6\wedge 2*x14+x5\wedge 2*x6*x14+x13*x14\wedge 2*\\{} & {} \quad x5+x13*x14\wedge 2*x6 +x13\wedge 2*x14*x5+x13\wedge 2*x14*x6)\\{} & {} \quad +B*(x5\wedge 2*x13*x14+x6\wedge 2*x13*x14)+C*(x5*x6*x13\wedge 2\\{} & {} \quad +x5*x6*x14\wedge 2)+D*(x5\wedge 2*x6\wedge 2 +x13\wedge 2*x14\wedge 2) \\{} & {} \quad +E*(x5\wedge 2*x14\wedge 2+x5\wedge 2*x13\wedge 2+x6\wedge 2*x14\wedge 2\\{} & {} \quad +x6\wedge 2*x13\wedge 2)+F*x5*x6*x13*x14;\\{} & {} >> f2=(x13*x14+x13*x17+x13*x18+x14*x17+x14*x18\\{} & {} \quad +x17*x18)\wedge 2;\\{} & {} >> f3=diff(diff(f1*f2,x13,2),x14,2)/factorial(2)\wedge 2; \\{} & {} \quad x13=0;x14=0; f3=expand(subs(f3)) \end{aligned}$$

$f3 =D*x6\wedge 2*x5\wedge 2+4*A*x17\wedge 2*x18*x5+4*A*x17*x18\wedge 2*x5+4*A*x17\wedge 2*x18*x6+4*A*x17*x18\wedge 2*x6 +2*C*x6*x5*x17\wedge 2+2*C*x6*x5*x18\wedge 2 +4*E*x5\wedge 2*x17*x18+4*E*x6\wedge 2*x17*x18+6*B*x5\wedge 2*x17*x18+6*B*x6\wedge 2*x17*x18+2*F*x5*x6*x17\wedge 2 +2*F*x5*x6*x18\wedge 2 +4*A*x6*x5\wedge 2*x17+4*A*x6*x5\wedge 2*x18+4*A*x6\wedge 2*x5*x17+4*A*x6\wedge 2*x5*x18+D*x17\wedge 2*x18\wedge 2 +2*E*x5\wedge 2*x17\wedge 2+6*F*x5*x6*x17*x18 +4*C*x6*x5*x17*x18+2*E*x5\wedge 2*x18\wedge 2+2*E*x6\wedge 2*x17\wedge 2+2*E*x6\wedge 2*x18\wedge 2 +2*B*x5\wedge 2*x17\wedge 2+2*B*x5\wedge 2*x18\wedge 2+2*B*x6\wedge 2*x17\wedge 2 +2*B*x6\wedge 2*x18\wedge 2$

$>>f4=-16*x6\wedge 2*x5*x18-16*x6*x5\wedge 2*x18-16*x6\wedge 2*x5*x17+44*x6*x5*x18\wedge 2+44*x6*x5*x17\wedge 2 +44*x17*x18*x6\wedge 2 +44*x17*x18*x5\wedge 2-16*x17*x18\wedge 2*x6-16*x17\wedge 2*x18*x6-16*x17*x18\wedge 2*x5-16*x17\wedge 2*x18*x5 -16*x6*x5\wedge 2*x17+16*x5\wedge 2*x17\wedge 2 +16*x5\wedge 2*x18\wedge 2+16*x6\wedge 2*x17\wedge 2+16*x6\wedge 2*x18\wedge 2+x17\wedge 2*x18\wedge 2 +120*x6*x5*x17*x18+x6\wedge 2*x5\wedge 2;$

$>>f5= diff(diff(diff(diff(f3*f4,x17,2),x18,2),x5,2),x6,2)/$

$factorial(2)\wedge 4$

$f5 =2*D-512*A+656*B+656*C+480*E+896*F$

Procedure 4

$$\begin{aligned}{} & {} >>syms\ \ x1\ \ x2\ \ x3\ \ x4\ \ x5\ \ x6\ \ x7\ \ x8\ \ x9\\{} & {} >>f1=(x1-x4)*(x1-x5)*(x1-x6)*(x1-x7)*(x1-x8)\\{} & {} \quad *(x1-x9);df1\\{} & {} \quad =diff(f1,x1,3)/factorial(3);\\{} & {} >>x1=0;df1=subs(df1);\\{} & {} >>f2=(x2-x4)*(x2-x5)*(x2-x6)*(x2-x7)*(x2-x8)\\{} & {} \quad *(x2-x9);df2\\{} & {} \quad =diff(f2,x2,3)/factorial(3);\\{} & {} >>x2=0;df2=subs(df2);\\{} & {} >>f3=(x3-x4)*(x3-x5)*(x3-x6)*(x3-x7)*(x3-x8)\\{} & {} \quad *(x3-x9);df3\\{} & {} \quad =diff(f3,x3,3)/factorial(3);\\{} & {} >>;x3=0;df3=subs(df3);\\{} & {} >>f4=df1*df2*df3*(x4-x7)*(x4-x8)*(x4-x9);df4\\{} & {} \quad =diff(f4,x4,3)/factorial(3);\\{} & {} >>x4=0;df4=subs(df4);\\{} & {} >>f5=df4*(x5-x7)*(x5-x8)*(x5-x9);df5=diff(f5,x5,3)\\{} & {} \quad /factorial(3);\\{} & {} >>x5=0;df5=subs(df5);\\{} & {} >>df5=expand(df5);\\{} & {} >>f6=df5*(x6-x7)*(x6-x8)*(x6-x9);\\{} & {} >>df61=diff(f6,x6,3)/factorial(3);x6=0;subs(df61) \end{aligned}$$

$ans=0$

$$\begin{aligned}{} & {} >>df62=diff(f6,x6,4)/factorial(4);x6=0;df62=subs(df62);\\{} & {} >>diff(diff(diff(df62,x7,3),x8,2),x9,3)/(factorial(3)\wedge 2\\{} & {} \quad *factorial(2)) \end{aligned}$$

$ans =-94$