Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more verti...Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.展开更多
Cockayne E J引入了一个图G的k-符号控制数γk-s11(G)的概念,提出了如下猜想:对任意n阶连通图G和正整数k(n2<k≤n),均有γk-s11(G)≤2k-n。我们证明了3方体Q3的5-符号控制数γ-5s11(Q3)=4,从而否定了这个猜想。此外,我们还给出了3-...Cockayne E J引入了一个图G的k-符号控制数γk-s11(G)的概念,提出了如下猜想:对任意n阶连通图G和正整数k(n2<k≤n),均有γk-s11(G)≤2k-n。我们证明了3方体Q3的5-符号控制数γ-5s11(Q3)=4,从而否定了这个猜想。此外,我们还给出了3-正则二部图k-符号控制数的一个上界,即证明了:对于任意n阶3-正则二部图G和正整数k(n2+1≤k≤n),均有γ-ks11(G)≤2(k+1)-n成立。展开更多
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination numb...Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.展开更多
Let <em>G</em>(<em>V</em>, <em>E</em>) be a finite connected simple graph with vertex set <em>V</em>(<em>G</em>). A function is a signed dominating function ...Let <em>G</em>(<em>V</em>, <em>E</em>) be a finite connected simple graph with vertex set <em>V</em>(<em>G</em>). A function is a signed dominating function <em>f </em>: <em style="white-space:normal;">V</em><span style="white-space:normal;">(</span><em style="white-space:normal;">G</em><span style="white-space:normal;">)</span><span style="white-space:nowrap;">→{<span style="white-space:nowrap;"><span style="white-space:nowrap;">−</span></span>1,1}</span> if for every vertex <em>v</em> <span style="white-space:nowrap;">∈</span> <em>V</em>(<em>G</em>), the sum of closed neighborhood weights of <em>v</em> is greater or equal to 1. The signed domination number <em>γ</em><sub>s</sub>(<em>G</em>) of <em>G</em> is the minimum weight of a signed dominating function on <em>G</em>. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths <em>P</em><sub><em>m</em></sub> and <em>P</em><sub><em>n</em></sub> for <em>m</em> = 6, 7 and arbitrary <em>n</em>.展开更多
文摘Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.
文摘Cockayne E J引入了一个图G的k-符号控制数γk-s11(G)的概念,提出了如下猜想:对任意n阶连通图G和正整数k(n2<k≤n),均有γk-s11(G)≤2k-n。我们证明了3方体Q3的5-符号控制数γ-5s11(Q3)=4,从而否定了这个猜想。此外,我们还给出了3-正则二部图k-符号控制数的一个上界,即证明了:对于任意n阶3-正则二部图G和正整数k(n2+1≤k≤n),均有γ-ks11(G)≤2(k+1)-n成立。
文摘Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.
文摘Let <em>G</em>(<em>V</em>, <em>E</em>) be a finite connected simple graph with vertex set <em>V</em>(<em>G</em>). A function is a signed dominating function <em>f </em>: <em style="white-space:normal;">V</em><span style="white-space:normal;">(</span><em style="white-space:normal;">G</em><span style="white-space:normal;">)</span><span style="white-space:nowrap;">→{<span style="white-space:nowrap;"><span style="white-space:nowrap;">−</span></span>1,1}</span> if for every vertex <em>v</em> <span style="white-space:nowrap;">∈</span> <em>V</em>(<em>G</em>), the sum of closed neighborhood weights of <em>v</em> is greater or equal to 1. The signed domination number <em>γ</em><sub>s</sub>(<em>G</em>) of <em>G</em> is the minimum weight of a signed dominating function on <em>G</em>. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths <em>P</em><sub><em>m</em></sub> and <em>P</em><sub><em>n</em></sub> for <em>m</em> = 6, 7 and arbitrary <em>n</em>.
