Let $G$ be a finite connected simple graph with a vertex set $V(G)$ and an edge set $E(G)$. A total signed dominating function of $G$ is a function $f: V(G)\cup E(G)\rightarrow \{-1, 1\}$, such that $\sum_{y\in N_T[x]}f(y) \geq 1$ for all $x\in V(G) \cup E(G)$. The total signed domination number of $G$ is the　minimum weight of a total signed dominating function on $G$. In this　paper, we prove lower and upper bounds on the total signed　domination number of the Cartesian product of two paths, $P_m\Box P_n$

Cao, Huiping

Gao, Hong

Yang, Yuansheng

University of Calgary Journal Hosting

On the total signed domination number of the Cartesian product of paths

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 12, Number 2, Pages 52–62ISSN 1715-0868ON THE TOTAL SIGNED DOMINATION NUMBER OFTHE CARTESIAN PRODUCT OF PATHSHONG GAO, QINGFANG ZHANG, AND YUANSHENG YANGAbstract. Let G be a finite connected simple graph with a vertex setV (G) and an edge set E(G). A total signed dominating function of Gis a function f : V (G) ∪ E(G)→ {−1, 1}, such that ∑y∈NT [x] f(y) ≥ 1for all x ∈ V (G) ∪ E(G). The total signed domination number of G isthe minimum weight of a total signed dominating function on G. In thispaper, we prove lower and upper bounds on the total signed dominationnumber of the Cartesian product of two paths, PmPn.1. IntroductionLet G be a finite connected simple graph with a vertex set V (G) andan edge set E(G). For v ∈ V (G), the open neighborhood of v is N(v) ={u | (u, v) ∈ E(G)}, and the closed neighborhood of v is N [v] = N(v) ∪{v}. For e ∈ E(G), the open neighborhood of e is N(e) = {g | g ∈E(G) is adjacent to e}, and the closed neighborhood of e is N [e] = N(e) ∪{e}. For an element x ∈ V (G)∪E(G), the total closed neighborhood of x isNT [x] = {y|y is adjacent to x or y is incident with x, y ∈ V (G)∪E(G)}∪{x}.We use [6] for terminology and notation which are not defined here.The fundamental concept concerning domination, namely the dominationnumber of a graph, was originally defined by means of a dominating set.This definition may be transferred into an equivalent definition done bymeans of a dominating function (the characteristic function of a dominatingset). A function f : V (G) → {0, 1} is called a domination function onG, if∑x∈N [v] f(x) ≥ 1 for each v ∈ V (G). The weight of f is w(f) =∑v∈V (G) f(v). The minimum of weights w(f), taken over all dominatingfunctions on G, is called the domination number γ(G) of G.The variations of the domination number may be obtained by replacingthe set {0, 1} by another set of numbers. If the closed interval [0, 1] on thereal line is taken instead of {0, 1}, then the fractional domination number isReceived by the editors October 9, 2014, and in revised form February 20, 2016.2000 Mathematics Subject Classification. Primary 05C15, 05C69.Key words and phrases. total signed domination number, Cartesian product, paths.This research was supported by the NSF of China (No.61562066), NSF of LiaoningProvince (No.2015020033), and FRF for the Central Universities(No.3132014324).c©2017 University of Calgary52TOTAL SIGNED DOMINATION NUMBER OF THE PRODUCT OF PATHS 53defined; by exchanging {0, 1} for {−1, 1}, the signed domination number isobtained.A signed dominating function is defined as f : V (G)→ {−1, 1} such that∑x∈N [v]f(x) ≥ 1 for all v ∈ V (G). The weight of f is w(f) =∑v∈V (G) f(v).The signed domination number γs(G) of G is the minimum weight of a signeddominating function on G.A total signed dominating function is defined as f : V (G) ∪ E(G) →{−1, 1}, such that F (x) = ∑y∈NT [x] f(y) ≥ 1 for all x ∈ V (G) ∪ E(G).The weight of f is w(f) =∑x∈V (G)∪E(G) f(x). The total signed dominationnumber γ∗s (G) of G is the minimum weight of a total signed dominatingfunction on G. In [3], Lu gave lower bounds for the total signed dominationnumber of a graph G and computed the exact values of γ∗s (Cn) and γ∗s (Pn)(n ≥ 3). In [4], Yuan and his collaborators studied the total signed domi-nation number of n ·Cm. Zou [7] gave the lower bounds on the total signeddomination number of some graphs.For two graphs G and H, the Cartesian product of G and H is the graphdenoted GH, where vi,j ∈ V (GH) if and only if vi ∈ V (G) and vj ∈V (H), and (vi1,j1 , vi2,j2) ∈ E(GH) if and only if i1 = i2 and (j1, j2) ∈ E(H)or j1 = j2 and (i1, i2) ∈ E(G). The study of domination numbers of productsof graphs was initiated by Vizing [5]. A survey and recent results on Vizing’sconjecture can be found in [1].In this paper, we study the total signed domination number of Carte-sian products of two paths. We prove a lower bound on the total signeddomination number of PmPn (m,n ≥ 2),γ∗s (PmPn) ≥⌈15mn− 3m− 3n− 4045⌉P(3mn−m−n).We then construct some total signed dominating functions and with them,present an upper bound of γ∗s (PmPn),γ∗s (PmPn) ≤mn+m+ n+ 22.The following are some important results on the total signed dominationnumber of Pn and the signed domination number of PmPn.Theorem 1.1. (Lu [3]) For any graph G,γ∗s (G) ≥⌈δ(G)−∆(G) + 1δ(G) + ∆(G) + 1(|E(G)|+ |V (G)|)⌉P(|E(G)|+|V (G)|)where P(s) is defined to be the parity of s, that is, P(s) = odd if s is oddand P(s) = even if s is even. Furthermore, this bound is sharp.Based on the Theorem 1.1, we can easily obtain the lower bounds for thetotal signed domination number γ∗s (PmPn).54 HONG GAO, QINGFANG ZHANG, AND YUANSHENG YANGCorollary 1.2. For any positive integers m,n ≥ 2,γ∗s (PmPn) ≥⌈−3mn−m− n7⌉P(3mn−m−n).Theorem 1.3 (Lu [3]). For n ≥ 3,γ∗s (Pn) =⌈2n−15⌉+ 1, if n (mod 5) ≡ 0 or 4,⌈2n−15⌉, if n (mod 5) ≡ 1 or 3,⌈2n−15⌉+ 2, if n (mod 5) ≡ 2.Theorem 1.4 (Haas [2]).γs(P2Pn) ={n, n even,n+ 1, n odd.For n ≥ 3,7n5− 85≤ γs(P3Pn) ≤ 7n5+ 2− 2(n (mod 5))5.For m,n ≥ 4,mn+ 4m+ 4n− 245≤ γs(PmPn) ≤ mn+ 8n+ 4m5.2. Lower bounds on the total signed domination number ofgraph PmPnIn this section, we prove that lower bounds on the total signed dominationnumber of PmPn (m,n ≥ 2) can be greater than zero (Corollary 1.2).Let G = PmPn with V (G) = {vi,j | 0 ≤ i ≤ m− 1, 0 ≤ j ≤ n− 1} andE(G) = {ei,j | ei,j = (vi,j , vi+1,j), 0 ≤ i ≤ m−1, 0 ≤ j ≤ n−2}∪{e′i,j | e′i,j =(vi,j , vi+1,j), 0 ≤ i ≤ m− 2, 0 ≤ j ≤ n− 1} (see Figure 1).Theorem 2.1. For any integers m,n ≥ 2,γ∗s (PmPn) ≥⌈15mn− 3m− 3n− 4045⌉P(3mn−m−n).Proof. Let f be an arbitrary total signed dominating function of graphG = PmPn. Then we have(2.1)∑y∈V (G)∪E(G)∑x∈NT [y]f(x) ≥ 3mn−m− n.TOTAL SIGNED DOMINATION NUMBER OF THE PRODUCT OF PATHS 55...............v0;0v1;0v2;0vn   2;0vn   1;0v0;1v1;1v2;1vn   2;1vn   1;1v0;2v1;2v2;2vn   2;2vn   1;2v0;m   2v1;m   2v2;m   2vn   2;m   2vn   1;m   2v0;m   1v1;m   1v2;m   1vn   2;m   1vn   1;m   1e0;0e1;0en   2;0e0;1e1;1en   2;1e0;2e1;2en   2;2e0;m   2e1;m   2en   2;m   2e0;m   1e1;m   1en   2;m   1e00;0e01;0e02;0e0n   2;0e0n   1;0e00;1e01;1e02;1e0n   2;1e0n   1;1e00;m   2e01;m   2e02;m   2e0n   2;m   2e0n   1;m   2Figure 1. Graph PmPn.Clearly for 0 ≤ i ≤ n−1, F (vi,0)−f(vi,1) ≥ 0 and F (vi,m−1)−f(vi,m−2) ≥ 0.Similarly for 0 ≤ i ≤ m−1, F (v0,i)−f(v1,i) ≥ 0 and F (vn−1,i)−f(vn−2,i) ≥0. Therefore the sum∑n−1i=0 (F (vi,0)− f(vi,1)) +∑n−1i=0 (F (vi,m−1)− f(vi,m−2))+∑m−1i=0 (F (v0,i)− f(v1,i)) +∑m−1i=0 (F (vn−1,i)− f(vn−2,i)) ≥ 0.Using the fact that f(x) ≥ −1 for all x ∈ V (G), we conclude(2.2)3∑n−1i=0 f(vi,0) + 2∑n−2i=0 f(ei,0) +∑n−1i=0 f(e′i,0)3∑n−1i=0 f(vi,m−1) + 2∑n−2i=0 f(ei,m−1) +∑n−1i=0 f(e′i,m−2)+3∑m−1i=0 f(v0,i) + 2∑m−2i=0 f(e′0,i) +∑m−1i=0 f(e0,i)+3∑m−1i=0 f(vn−1,i) + 2∑m−2i=0 f(e′n−1,i) +∑m−1i=0 f(en−1,i)≥ −8.Note that since F (v0,0) ≥ 1, f(v0,0) + f(e′0,0) + f(e0,0) ≥ −1. Analogously,we have f(vn−1,0) + f(e′n−1,0) + f(en−2,0) ≥ −1, f(v0,m−1) + f(e′0,m−1) +f(e0,m−1) ≥ −1 and f(vn−1,m−1) + f(e′n−1,m−1) + f(en−2,m−1) ≥ −1. SinceF (x) ≥ 1 for all x,n−1∑i=0F (ei,0) +n−1∑i=0F (ei,m−1) +m−1∑i=0F (e′0,i) +m−1∑i=0F (e′n−1,i) ≥ 2m+ 2n− 4.It follows that(2.3)2∑n−1i=0 f(vi,0) + 3∑n−2i=0 f(ei,0) + 2∑n−1i=0 f(e′i,0)+2∑n−1i=0 f(vi,m−1) + 3∑n−2i=0 f(ei,m−1) + 2∑n−1i=0 f(e′i,m−2)+2∑m−1i=0 f(v0,i) + 3∑m−2i=0 f(e′0,i) + 2∑m−1i=0 f(e0,i)+2∑m−1i=0 f(vn−1,i) + 3∑m−2i=0 f(e′n−1,i) + 2∑m−1i=0 f(en−2,i)≥ 2m+ 2n− 12.56 HONG GAO, QINGFANG ZHANG, AND YUANSHENG YANGAdding equation (2.2) and (2.3),5∑n−1i=0 f(vi,0) + 5∑n−2i=0 f(ei,0) + 3∑n−1i=0 f(e′i,0)+5∑n−1i=0 f(vi,m−1) + 5∑n−2i=0 f(ei,m−1) + 3∑n−1i=0 f(e′i,m−2)+5∑m−1i=0 f(v0,i) + 5∑m−2i=0 f(e′0,i) + 3∑m−1i=0 f(e0,i)+5∑m−1i=0 f(vn−1,i) + 5∑m−2i=0 f(e′n−1,i) + 3∑m−1i=0 f(en−2,i)≥ 2n+ 2m− 20.This implies(2.4)10∑n−1i=0 f(vi,0) + 10∑n−2i=0 f(ei,0) + 5∑n−1i=0 f(e′i,0)+10∑n−1i=0 f(vi,m−1) + 10∑n−2i=0 f(ei,m−1) + 5∑n−1i=0 f(e′i,m−2)+10∑m−1i=0 f(v0,i) + 10∑m−2i=0 f(e′0,i) + 5∑m−1i=0 f(e0,i)+10∑m−1i=0 f(vn−1,i) + 10∑m−2i=0 f(e′n−1,i) + 5∑m−1i=0 f(en−2,i)≥ 4m+ 4n− 40−∑n−1i=0 f(e′i,0)−∑n−1i=0 f(e′i,m−2)−∑m−1i=0 f(e0,i)−∑m−1i=0 f(en−2,i)≥ 2m+ 2n− 40.Finally observe,(2.5)∑y∈V (G)∪E(G)∑x∈NT [y] f(x)= 9∑y∈V (G)∪E(G) f(y)− 2∑n−1i=0 f(vi,0)− 2∑n−2i=0 f(ei,0)−∑n−1i=0 f(e′i,0)− 2∑n−1i=0 f(vi,m−1)− 2∑n−2i=0 f(ei,m−1)−∑n−1i=0 f(e′i,m−2)− 2∑m−1i=0 f(v0,i)− 2∑m−2i=0 f(e′0,i)−∑m−1i=0 f(e0,i)− 2∑m−1i=0 f(vn−1,i)− 2∑m−2i=0 f(e′n−1,i)−∑m−1i=0 f(en−2,i).From equations (2.1),(2.4), and (2.5) it follows that45∑y∈V (G)∪E(G) f(y)= 5∑y∈V (G)∪E(G)∑x∈NT [y] f(x)+10∑n−1i=0 f(vi,0) + 10∑n−2i=0 f(ei,0) + 5∑n−1i=0 f(e′i,0)+10∑n−1i=0 f(vi,m−1) + 10∑n−2i=0 f(ei,m−1) + 5∑n−1i=0 f(e′i,m−2)+10∑m−1i=0 f(v0,i) + 10∑m−2i=0 f(e′0,i) + 5∑m−1i=0 f(e0,i)+10∑m−1i=0 f(vn−1,i) + 10∑m−2i=0 f(e′n−1,i) + 5∑m−1i=0 f(en−2,i)≥ 5(3mn−m− n) + 2m+ 2n− 40 = 15mn− 3m− 3n− 40.Henceγ∗s (PmPn) ≥⌈15mn−3m−3n−4045⌉P(3mn−m−n).3. Upper bounds on the total signed domination number ofgraph PmPnIn this section, we present upper bounds on the total signed dominationnumber of PmPn for m,n ≥ 2. We introduce the following notation toTOTAL SIGNED DOMINATION NUMBER OF THE PRODUCT OF PATHS 57define a total signed dominating function of PmPn,f =f(v0,0) f(e′0,0) f(v0,1) f(e′0,1) f(v0,2) · · · f(v0,m−2) f(e′0,m−2) f(v0,m−1)f(e0,0) f(e0,1) f(e0,2) · · · f(e0,m−2) f(e0,m−1)f(v1,0) f(e′1,0) f(v1,1) f(e′1,1) f(v1,2) · · · f(v1,m−2) f(e′1,m−2) f(v1,m−1)f(e1,0) f(e1,1) f(e1,2) · · · f(e1,m−2) f(e1,m−1)...............f(vn−2,0)f(e′n−2,0)f(vn−2,1)f(e′n−2,1)f(vn−2,2)· · · f(vn−2,m−2)f(e′n−2,m−2)f(vn−2,m−1)f(en−2,0) f(en−2,1) f(en−2,2)· · · f(en−2,m−2) f(en−2,m−1)f(vn−1,0)f(e′n−1,0)f(vn−1,1)f(e′n−1,1)f(vn−1,2)· · · f(vn−1,m−2)f(e′n−1,m−2)f(vn−1,m−1) .Lemma 3.1. For m ≥ 2 and n = 2,γ∗s (PmPn) ≤ m.Proof. It is sufficient to define a function f with w(f) = m. Letf =(−1 1 · · · −1 1 −11 · · · 1 11−1 · · · 1−1 1).P32P2Figure 2. Graphs P3P2 corresponding to f .By adding each column, one can see w(f) = m. For m = 3 and n = 2, seeFigure 2, where black vertices (thick edges) stand for f(x) = 1, and whitevertices (thin edges) stand for f(x) = −1. Lemma 3.2. For m ≥ 2 and n ≥ 3,γ∗s (PmPn) ≤mn+m+42 , m (mod 4) ≡ 0 and n (mod 4) ≡ 0,mn+m+22 , m (mod 4) ≡ 2 or n (mod 4) ≡ 2,mn+m−22 , m (mod 2) ≡ 0 and n (mod 2) ≡ 1,mn+m+n−52 , m (mod 2) ≡ 1 and n (mod 2) ≡ 1.Proof.Case 1 : m (mod 2) ≡ 0 and n (mod 2) ≡ 0.Subcase 1.1. m (mod 4) ≡ 0 and n (mod 4) ≡ 0.58 HONG GAO, QINGFANG ZHANG, AND YUANSHENG YANGIt is sufficient to define a function f with w(f) = (mn+m+ 4)/2.Letf =1−1 1 1−1 1−1 1 · · · 1−1 1 1−1 1−1 1 1−1 1 1−1 1 11 1 −1 1 · · · 1 1 −1 1 1 1 −1 1−1 1−1 1 1−1 1−1 · · · −1 1−1 1 1−1 1−1 −1 1−1 1 1−1 11 −1 1 1 · · · 1 −1 1 1 1 −1 1 −1−1 1−1 1 1−1 1−1 · · · −1 1−1 1 1−1 1−1 −1 1−1 1 1−1 11 1 −1 1 · · · 1 1 −1 1 1 1 −1 11−1 1 1−1 1−1 1 · · · 1−1 1 1−1 1−1 1 1−1 1 1−1 1−1−1 −1 1 −1 · · · −1 −1 1 −1 −1 −1 1 11−1 1 1−1 1−1 1 · · · 1−1 1 1−1 1−1 1 1−1 1 1−1 1−11 1 −1 1 · · · 1 1 −1 1 1 1 −1 1....................................−1 1−1 1 1−1 1−1 · · · −1 1−1 1 1−1 1−1 −1 1−1 1 1−1 11 −1 1 1 · · · 1 −1 1 1 1 −1 1 −1−1 1−1 1 1−1 1−1 · · · −1 1−1 1 1−1 1−1 −1 1−1 1 1−1 11 1 −1 1 · · · 1 1 −1 1 1 1 −1 11−1 1 1−1 1−1 1 · · · 1−1 1 1−1 1−1 1 1−1 1 1−1 1−1−1 −1 1 −1 · · · −1 −1 1 −1 −1 −1 1 11−1 1 1−1 1−1 1 · · · 1−1 1 1−1 1−1 1 1−1 1 1−1 1−11 1 −1 1 · · · 1 1 −1 1 1 1 −1 1−1 1−1 1 1−1 1−1 · · · −1 1−1 1 1−1 1−1 −1 1−1 1 1−1 11 −1 1 1 · · · 1 −1 1 1 1 −1 1 −1−1 1−1 1 1−1 1−1 · · · −1 1−1 1 1−1 1−1 −1 1−1 1 1−1 11 1 −1 1 · · · 1 1 −1 1 1 1 −1 11−1 1 1−1 1−1 1 · · · 1−1 1 1−1 1−1 1 1−1 1 1−1 1 1.By adding each column, one can see w(f) = (mn + m + 4)/2 (seeFigure 3 for m = 12 and n = 12).P122P12Figure 3. Graph P12P12 corresponding to f .Subcase 1.2. m (mod 4) ≡ 2 or n (mod 4) ≡ 2.TOTAL SIGNED DOMINATION NUMBER OF THE PRODUCT OF PATHS 59It is sufficient to define a function f with w(f) = (mn+m+ 2)/2.Letf =−1 1−1 1−1 1−1 1 · · · −1 1−1 1−1 1−1 1 −1 1−11 1 1 1 · · · 1 1 1 1 1 11−1 1−1 1 1 1−1 · · · 1−1 1−1 1 1 1−1 1−1 11 1 −1 −1 · · · 1 1 −1 −1 1 1−1−1−1 1−1 1−1 1 · · · −1−1−1 1−1 1−1 1 −1−1−11 1 1 1 · · · 1 1 1 1 1 1..............................1−1 1−1 1 1 1−1 · · · 1−1 1−1 1 1 1−1 1−1 11 1 −1 −1 · · · 1 1 −1 −1 1 1−1−1−1 1−1 1−1 1 · · · −1−1−1 1−1 1−1 1 −1−1−11 1 1 1 · · · 1 1 1 1 1 11−1 1−1 1 1 1−1 · · · 1−1 1−1 1 1 1−1 1−1 11 1 −1 −1 · · · 1 1 −1 −1 1 11−1−1 1−1 1 1 1 · · · 1−1−1 1−1 1 1 1 1−1−1−1 1 1 −1 · · · −1 1 1 −1 −1 11 1−1 1−1 1 1−1 · · · 1 1−1 1−1 1 1−1 1 1 1.By adding each column, one can see w(f) = (mn + m + 2)/2 (seeFigure 4 for m = 10 and n = 8).P102P8Figure 4. Graphs P10P8 corresponding to f .Case 2 : m (mod 2) ≡ 0 and n (mod 2) ≡ 1.It is sufficient to define a function f with w(f) = (mn+m−2)/2. Letf =−1 1−1 1 · · · −1 1−1 1 −1 1−11 1 · · · 1 1 1 11−1 1−1 · · · 1−1 1−1 1−1 11 1 · · · 1 1 1 1−1−1−1 1 · · · −1−1−1 1 −1−1−11 1 · · · 1 1 1 1..................1−1 1−1 · · · 1−1 1−1 1−1 11 1 · · · 1 1 1 1−1−1−1 1 · · · −1−1−1 1 −1−1−11 1 · · · 1 1 1 11−1 1−1 · · · 1−1 1−1 1−1 11 1 · · · 1 1 1 1−1 1−1 1 · · · −1 1−1 1 −1 1−1.60 HONG GAO, QINGFANG ZHANG, AND YUANSHENG YANGBy adding each column, one can see w(f) = (mn+m− 2)/2 (see Figure5 for m = 8 and n = 7).P82P7Figure 5. Graphs P8P7 corresponding to f .Case 3 : m (mod 2) ≡ 1 and n (mod 2) ≡ 1.It is sufficient to define a function f with w(f) = (mn+m−5)/2. Letf =−1 1−1 1 · · · −1 1−1 1 −11 1 · · · 1 1 11−1 1−1 · · · 1−1 1−1 11 1 · · · 1 1 1−1−1−1 1 · · · −1−1−1 1 −11 1 · · · 1 1 1...............1−1 1−1 · · · 1−1 1−1 11 1 · · · 1 1 1−1−1−1 1 · · · −1−1−1 1 −11 1 · · · 1 1 11−1 1−1 · · · 1−1 1−1 11 1 · · · 1 1 1−1 1−1 1 · · · −1 1−1 1 −1.P52P7Figure 6. Graphs P5P7 corresponding to f .TOTAL SIGNED DOMINATION NUMBER OF THE PRODUCT OF PATHS 61By adding each column, one can see w(f) = (mn + m − 5)/2 (seeFigure 6 for m = 5 and n = 7).By Lemma 3.1 and Lemma 3.2, we haveTheorem 3.3. For any integers m,n ≥ 2,γ∗s (PmPn) ≤m, n = 2,mn+m+42 , n ≥ 3 and m (mod 4) ≡ 0 and n (mod 4) ≡ 0,mn+m+22 , n ≥ 3 and m (mod 4) ≡ 2 or n (mod 4) ≡ 2,mn+m−22 , n ≥ 3 and m (mod 2) ≡ 0 and n (mod 2) ≡ 1,mn+m+n−52 , n ≥ 3 and m (mod 2) ≡ 1 and n (mod 2) ≡ 1.that is,γ∗s (PmPn) ≤mn+m+ n+ 22.By Theorem 2.1 and Theorem 3.3, we haveTheorem 3.4. For any integers m,n ≥ 2,⌈15mn− 3m− 3n− 4045⌉P(3mn−m−n)≤ γ∗s (PmPn) ≤mn+m+ n+ 22.AcknowledgementThe authors are grateful to the referees for their valuable comments andsuggestionsReferences1. B. Bresar, P. Dorbec, W. Goddard, B.L. Hartnell, M.A. Henning, S. Klavzar, and D.F.Rall, Vizing’s conjecture: a survey and recent results, Journal of Graph Theory 69(2011), no. 1, 46–76.2. R. Haas and T.B. Wexler, Bounds on the signed domination number of a graph, Elec-tronic Notes in Discrete Mathematics 11 (2002), 742–750.3. X.Z. Lu, A lower bound on the total signed domination numbers of graphs, Science inChina Series A: Mathematics 50 (2007), no. 8, 1157–1162.4. Y. Ren, W. Feng, and Jirimutu, On the total signed domination number of n · Cm,International Journal of Pure and Applied Mathematics 81 (2012), no. 5, 765–772.5. V.G. Vizing, The Cartesian product of graphs, Vycisl. Sistemy 9 (1963), 30–43.6. D.B. West, Introduction to graph theory, Prentice-Hall, Inc (2000).7. X.L. Zou, X.G. Chen, and G.Y. Sun, Lower bounds on the total signed dominationnumber of graphs, Applied Mathematical Sciences 1 (2007), no. 50, 2499–2504.62 HONG GAO, QINGFANG ZHANG, AND YUANSHENG YANGDepartment of Mathematics, Dalian Maritime University Dalian,China 116026E-mail address: gaohong@dlmu.edu.cnDepartment of Mathematics, Dalian Maritime University Dalian,China 116026E-mail address: gao hong@qq.comSchool of Computer Science and Technology, Dalian University ofTechnology, Dalian, China 116024E-mail address: yangys@dlut.edu.cn

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On the total signed domination number of the Cartesian product of paths

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