Equality in a bound that relates the size and the restrained domination number of a graph

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a restrained dominating set if every vertex in \(V-S\) is adjacent to a vertex in \(S\) and to a vertex in \(V-S\). The restrained domination number of \(G\), denoted \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). The best possible upper bound \(q(n,k)\) is established in Joubert (Discrete Appl Math 161:829–837, 2013) on the size \(m(G)\) of a graph \(G\) with a given order \(n \ge 5\) and restrained domination number \(k \in \{3, \ldots , n-2\}\). We extend this result to include the cases \(k=1,2,n\), and characterize graphs \(G\) of order \(n \ge 1\) and restrained domination number \(k \in \{1,\dots , n-2,n\}\) for which \(m(G)=q(n,k)\).     

Signed total Roman domination in graphs

Let \(G\) be a finite and simple graph with vertex set \(V(G)\). A signed total Roman dominating function (STRDF) on a graph \(G\) is a function \(f:V(G)\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{x\in N(v)}f(x)\ge 1\) for each vertex \(v\in V(G)\), where \(N(v)\) is the neighborhood of \(v\), and (ii) every vertex \(u\) for which \(f(u)=-1\) is adjacent to at least one vertex \(v\) for which \(f(v)=2\). The weight of an SRTDF \(f\) is \(\sum _{v\in V(G)}f(v)\). The signed total Roman domination number \(\gamma _{stR}(G)\) of \(G\) is the minimum weight of an STRDF on \(G\). In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on \(\gamma _{stR}(G)\). In addition, we determine the signed total Roman domination number of some classes of graphs.     

Perfect graphs involving semitotal and semipaired domination

Let G be a graph with vertex set V and no isolated vertices, and let S be a dominating set of V. The set S is a semitotal dominating set of G if every vertex in S is within distance 2 of another vertex of S. And, S is a semipaired dominating set of G if S can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semitotal domination number \(\gamma _\mathrm{t2}(G)\) is the minimum cardinality of a semitotal dominating set of G, and the semipaired domination number \(\gamma _\mathrm{pr2}(G)\) is the minimum cardinality of a semipaired dominating set of G. For a graph without isolated vertices, the domination number \(\gamma (G)\), the total domination \(\gamma _t(G)\), and the paired domination number \(\gamma _\mathrm{pr}(G)\) are related to the semitotal and semipaired domination numbers by the following inequalities: \(\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _t(G) \le \gamma _\mathrm{pr}(G)\) and \(\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _\mathrm{pr2}(G) \le \gamma _\mathrm{pr}(G) \le 2\gamma (G)\). Given two graph parameters \(\mu \) and \(\psi \) related by a simple inequality \(\mu (G) \le \psi (G)\) for every graph G having no isolated vertices, a graph is \((\mu ,\psi )\)-perfect if every induced subgraph H with no isolated vertices satisfies \(\mu (H) = \psi (H)\). Alvarado et al. (Discrete Math 338:1424–1431, 2015) consider classes of \((\mu ,\psi )\)-perfect graphs, where \(\mu \) and \(\psi \) are domination parameters including \(\gamma \), \(\gamma _t\) and \(\gamma _\mathrm{pr}\). We study classes of perfect graphs for the possible combinations of parameters in the inequalities when \(\gamma _\mathrm{t2}\) and \(\gamma _\mathrm{pr2}\) are included in the mix. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs.     

Disjunctive total domination in graphs

Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, \(\gamma _t(G)\). A set \(S\) of vertices in \(G\) is a disjunctive total dominating set of \(G\) if every vertex is adjacent to a vertex of \(S\) or has at least two vertices in \(S\) at distance \(2\) from it. The disjunctive total domination number, \(\gamma ^d_t(G)\), is the minimum cardinality of such a set. We observe that \(\gamma ^d_t(G) \le \gamma _t(G)\). We prove that if \(G\) is a connected graph of order \(n \ge 8\), then \(\gamma ^d_t(G) \le 2(n-1)/3\) and we characterize the extremal graphs. It is known that if \(G\) is a connected claw-free graph of order \(n\), then \(\gamma _t(G) \le 2n/3\) and this upper bound is tight for arbitrarily large \(n\). We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if \(G\) is a connected claw-free graph of order \(n > 14\), then \(\gamma ^d_t(G) \le 4n/7\) and we characterize the graphs achieving equality in this bound.     

A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Let \(G=(V,E)\) be a graph and \(\phi \) be a total \(k\)-coloring of \(G\) using the color set \(\{1,\ldots , k\}\). Let \(\sum _\phi (u)\) denote the sum of the color of the vertex \(u\) and the colors of all incident edges of \(u\). A \(k\)-neighbor sum distinguishing total coloring of \(G\) is a total \(k\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(\sum _\phi (u)\ne \sum _\phi (v)\). By \(\chi ^{''}_{nsd}(G)\), we denote the smallest value \(k\) in such a coloring of \(G\). Pil?niak and Wo?niak first introduced this coloring and conjectured that \(\chi _{nsd}^{''}(G)\le \Delta (G)+3\) for any simple graph \(G\). In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with \(\Delta (G)\ge 7\). Moreover, we also show that \(\chi _{nsd}^{''}(G)\le \Delta (G)+2\) for planar graphs without intersecting triangles with \(\Delta (G) \ge 9\). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.     

Reconfiguration of dominating sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of \(D_k(G)\), the graph consisting of a node for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that \(D_{\varGamma (G)+1}(G)\) is not necessarily connected, for \(\varGamma (G)\) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for \(b \ge 3\). Moreover, we construct an infinite family of graphs such that \(D_{\gamma (G)+1}(G)\) has exponential diameter, for \(\gamma (G)\) the minimum size of a dominating set. On the positive side, we show that \(D_{n-\mu }(G)\) is connected and of linear diameter for any graph G on n vertices with a matching of size at least \(\mu +1\).     

The 2-surviving rate of planar graphs without 5-cycles

Let \(G\) be a connected graph with \(n\ge 2\) vertices. Let \(k\ge 1\) be an integer. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects \(k\)-vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let \(\hbox {sn}_k(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The \(k\)-surviving rate \(\rho _k(G)\) of \(G\) is defined to be \(\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)\), which is the average proportion of saved vertices. In this paper, we prove that if \(G\) is a planar graph with \(n\ge 2\) vertices and without 5-cycles, then \(\rho _2(G)>\frac{1}{363}\).     

Algorithmic aspects of <Emphasis Type="Italic">b</Emphasis>-disjunctive domination in graphs

For a fixed integer \(b>1\), a set \(D\subseteq V\) is called a b-disjunctive dominating set of the graph \(G=(V,E)\) if for every vertex \(v\in V{\setminus }D\), v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The Minimum b-Disjunctive Domination Problem (MbDDP) is to find a b-disjunctive dominating set of minimum cardinality. The cardinality of a minimum b-disjunctive dominating set of G is called the b-disjunctive domination number of G, and is denoted by \(\gamma _{b}^{d}(G)\). Given a positive integer k and a graph G, the b-Disjunctive Domination Decision Problem (bDDDP) is to decide whether G has a b-disjunctive dominating set of cardinality at most k. In this paper, we first show that for a proper interval graph G, \(\gamma _{b}^{d}(G)\) is equal to \(\gamma (G)\), the domination number of G for \(b \ge 3\) and observe that \(\gamma _{b}^{d}(G)\) need not be equal to \(\gamma (G)\) for \(b=2\). We then propose a polynomial time algorithm to compute a minimum cardinality b-disjunctive dominating set of a proper interval graph for \(b=2\). Next we tighten the NP-completeness of bDDDP by showing that it remains NP-complete even in chordal graphs. We also propose a \((\ln ({\varDelta }^{2}+(b-1){\varDelta }+b)+1)\)-approximation algorithm for MbDDP, where \({\varDelta }\) is the maximum degree of input graph \(G=(V,E)\) and prove that MbDDP cannot be approximated within \((1-\epsilon ) \ln (|V|)\) for any \(\epsilon >0\) unless NP \(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we show that MbDDP is APX-complete for bipartite graphs with maximum degree \(\max \{b,4\}\).     

Signed mixed Roman domination numbers in graphs

Let \(G = (V;E)\) be a simple graph with vertex set \(V\) and edge set \(E\). A signed mixed Roman dominating function (SMRDF) of \(G\) is a function \(f: V\cup E\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{y\in N_m[x]}f(y)\ge 1\) for each \(x\in V\cup E\), where \(N_m[x]\) is the set, called mixed closed neighborhood of \(x\), consists of \(x\) and the elements of \(V\cup E\) adjacent or incident to \(x\) (ii) every element \(x\in V\cup E\) for which \(f(x) = -1\) is adjacent or incident to at least one element \(y\in V\cup E\) for which \(f(y) = 2\). The weight of a SMRDF \(f\) is \(\omega (f)=\sum _{x\in V\cup E}f(x)\). The signed mixed Roman domination number \(\gamma _{sR}^*(G)\) of \(G\) is the minimum weight of a SMRDF of \(G\). In this paper we initiate the study of the signed mixed Roman domination number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.     

Cacti with the smallest,second smallest,and third smallest Gutman index

The Gutman index (also known as Schultz index of the second kind) of a graph \(G\) is defined as \(Gut(G)=\sum \nolimits _{u,v\in V(G)}d(u)d(v)d(u, v)\). A graph \(G\) is called a cactus if each block of \(G\) is either an edge or a cycle. Denote by \(\mathcal {C}(n, k)\) the set of connected cacti possessing \(n\) vertices and \(k\) cycles. In this paper, we give the first three smallest Gutman indices among graphs in \(\mathcal {C}(n, k)\), the corresponding extremal graphs are characterized as well.     

2-Edge connected dominating sets and 2-Connected dominating sets of a graph

A \(k\)-connected (resp. \(k\)-edge connected) dominating set \(D\) of a connected graph \(G\) is a subset of \(V(G)\) such that \(G[D]\) is \(k\)-connected (resp. \(k\)-edge connected) and each \(v\in V(G)\backslash D\) has at least one neighbor in \(D\). The \(k\) -connected domination number (resp. \(k\) -edge connected domination number) of a graph \(G\) is the minimum size of a \(k\)-connected (resp. \(k\)-edge connected) dominating set of \(G\), and denoted by \(\gamma _k(G)\) (resp. \(\gamma '_k(G)\)). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph \(G\), if it is \(2\)-edge connected, then \(\gamma '_2(G)\le 4\alpha (G)-1\), and it is \(2\)-connected, then \(\gamma _2(G)\le 6\alpha (G)-3\), where \(\alpha (G)\) is the independent number of \(G\).     

Neighbor sum distinguishing total coloring of graphs with bounded treewidth

A proper total k-coloring \(\phi \) of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\dots , k\}\) such that no adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let \(m_{\phi }(v)\) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if \(m_{\phi }(u)\not =m_{\phi }(v)\) for each edge \(uv\in E(G).\) Let \(\chi _{\Sigma }^t(G)\) be the neighbor sum distinguishing total chromatic number of a graph G. Pil?niak and Wo?niak conjectured that for any graph G, \(\chi _{\Sigma }^t(G)\le \Delta (G)+3\). In this paper, we show that if G is a graph with treewidth \(\ell \ge 3\) and \(\Delta (G)\ge 2\ell +3\), then \(\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1\). This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when \(\ell =3\) and \(\Delta \ge 9\), we show that \(\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2\) and characterize graphs with equalities.     

The 3-rainbow index and connected dominating sets

A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of \(G\) is called a 3-rainbow coloring if for any three vertices in \(G\), there exists a rainbow tree connecting them. The 3-rainbow index \(rx_3(G)\) of \(G\) is defined as the minimum number of colors that are needed in a 3-rainbow coloring of \(G\). This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected 3-way dominating sets and 3-dominating sets. We show that for every connected graph \(G\) on \(n\) vertices with minimum degree at least \(\delta \, (3\le \delta \le 5)\), \(rx_{3}(G)\le \frac{3n}{\delta +1}+4\), and the bound is tight up to an additive constant; whereas for every connected graph \(G\) on \(n\) vertices with minimum degree at least \(\delta \, (\delta \ge 3)\), we get that \(rx_{3}(G)\le \frac{\ln (\delta +1)}{\delta +1}(1+o_{\delta }(1))n+5\). In addition, we obtain some tight upper bounds of the 3-rainbow index for some special graph classes, including threshold graphs, chain graphs and interval graphs.     

The game chromatic index of some trees with maximum degree four and adjacent degree-four vertices

A (proper) total-k-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}\) such that any two adjacent or incident elements in \(V (G) \cup E(G)\) receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. An adjacent vertex distinguishing total-k-coloring of G is a total-k-coloring of G such that for each edge \(uv\in E(G)\), \(C(u)\ne C(v)\). We denote the smallest value k in such a coloring of G by \(\chi ^{\prime \prime }_{a}(G)\). It is known that \(\chi _{a}^{\prime \prime }(G)\le \Delta (G)+3\) for any planar graph with \(\Delta (G)\ge 10\). In this paper, we consider the list version of this coloring and show that if G is a planar graph with \(\Delta (G)\ge 11\), then \({ ch}_{a}^{\prime \prime }(G)\le \Delta (G)+3\), where \({ ch}^{\prime \prime }_a(G)\) is the adjacent vertex distinguishing total choosability.     

On the difference of two generalized connectivities of a graph

The concept of k-connectivity \(\kappa '_{k}(G)\) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity \(\kappa _{k}(G)\), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if \(\kappa '_k(G)\ne n-k+1\) where \(k\ge 3\), then \(0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1\); otherwise, \(-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k\). Moreover, all of these bounds are sharp. Some specific study is focused for the case \(k=3\). As results, we characterize the graphs with \(\kappa '_3(G)=\kappa _3(G)=t\) for \(t\in \{1, n-3, n-2\}\), and give a necessary condition for \(\kappa '_3(G)=\kappa _3(G)\) by showing that for a connected graph G of order n and size m, if \(\kappa '_3(G)=\kappa _3(G)=t\) where \(1\le t\le n-3\), then \(m\le {n-2\atopwithdelims ()2}+2t\). Moreover, the unique extremal graph is given for the equality to hold.     

On (s,t)-relaxed L(2,1)-labeling of graphs

We initiate the study of relaxed \(L(2,1)\)-labelings of graphs. Suppose \(G\) is a graph. Let \(u\) be a vertex of \(G\). A vertex \(v\) is called an \(i\)-neighbor of \(u\) if \(d_G(u,v)=i\). A \(1\)-neighbor of \(u\) is simply called a neighbor of \(u\). Let \(s\) and \(t\) be two nonnegative integers. Suppose \(f\) is an assignment of nonnegative integers to the vertices of \(G\). If the following three conditions are satisfied, then \(f\) is called an \((s,t)\)-relaxed \(L(2,1)\)-labeling of \(G\): (1) for any two adjacent vertices \(u\) and \(v\) of \(G, f(u)\not =f(v)\); (2) for any vertex \(u\) of \(G\), there are at most \(s\) neighbors of \(u\) receiving labels from \(\{f(u)-1,f(u)+1\}\); (3) for any vertex \(u\) of \(G\), the number of \(2\)-neighbors of \(u\) assigned the label \(f(u)\) is at most \(t\). The minimum span of \((s,t)\)-relaxed \(L(2,1)\)-labelings of \(G\) is called the \((s,t)\)-relaxed \(L(2,1)\)-labeling number of \(G\), denoted by \(\lambda ^{s,t}_{2,1}(G)\). It is clear that \(\lambda ^{0,0}_{2,1}(G)\) is the so called \(L(2,1)\)-labeling number of \(G\). \(\lambda ^{1,0}_{2,1}(G)\) is simply written as \(\widetilde{\lambda }(G)\). This paper discusses basic properties of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of graphs. For any two nonnegative integers \(s\) and \(t\), the exact values of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of paths, cycles and complete graphs are determined. Tight upper and lower bounds for \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of complete multipartite graphs and trees are given. The upper bounds for \((s,1)\)-relaxed \(L(2,1)\)-labeling number of general graphs are also investigated. We introduce a new graph parameter called the breaking path covering number of a graph. A breaking path \(P\) is a vertex sequence \(v_1,v_2,\ldots ,v_k\) in which each \(v_i\) is adjacent to at least one vertex of \(v_{i-1}\) and \(v_{i+1}\) for \(i=2,3,\ldots ,k-1\). A breaking path covering of \(G\) is a set of disjoint such vertex sequences that cover all vertices of \(G\). The breaking path covering number of \(G\), denoted by \(bpc(G)\), is the minimum number of breaking paths in a breaking path covering of \(G\). In this paper, it is proved that \(\widetilde{\lambda }(G)= n+bpc(G^{c})-2\) if \(bpc(G^{c})\ge 2\) and \(\widetilde{\lambda }(G)\le n-1\) if and only if \(bpc(G^{c})=1\). The breaking path covering number of a graph is proved to be computable in polynomial time. Thus, if a graph \(G\) is of diameter two, then \(\widetilde{\lambda }(G)\) can be determined in polynomial time. Several conjectures and problems on relaxed \(L(2,1)\)-labelings are also proposed.     

Neighbor sum distinguishing list total coloring of subcubic graphs

Let \(G=(V, E)\) be a simple graph and denote the set of edges incident to a vertex v by E(v). The neighbor sum distinguishing (NSD) total choice number of G, denoted by \(\mathrm{ch}_{\Sigma }^{t}(G)\), is the smallest integer k such that, after assigning each \(z\in V\cup E\) a set L(z) of k real numbers, G has a total coloring \(\phi \) satisfying \(\phi (z)\in L(z)\) for each \(z\in V\cup E\) and \(\sum _{z\in E(u)\cup \{u\}}\phi (z)\ne \sum _{z\in E(v)\cup \{v\}}\phi (z)\) for each \(uv\in E\). In this paper, we propose some reducible configurations of NSD list total coloring for general graphs by applying the Combinatorial Nullstellensatz. As an application, we present that \(\mathrm{ch}^{t}_{\Sigma }(G)\le \Delta (G)+3\) for every subcubic graph G.     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).     

Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph \(G=(V_G, E_G)\) is defined as \(\bar{R}(G)=\sum _{u,v \in V_G}(\delta _G(u)+\delta _G(v))\frac{1}{d_G(u,v)}\), where \(\delta _G(u)\) is the vertex degree of \(u\), and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). The reciprocal degree distance is an additive weight version of the Harary index, which is defined as \(H(G)=\sum _{u,v \in V_G}\frac{1}{d_G(u,v)}\). In this paper, the extremal \(\bar{R}\)-values on several types of important graphs are considered. The graph with the maximum \(\bar{R}\)-value among all the simple connected graphs of diameter \(d\) is determined. Among the connected bipartite graphs of order \(n\), the graph with a given matching number (resp. vertex connectivity) having the maximum \(\bar{R}\)-value is characterized. Finally, sharp upper bounds on \(\bar{R}\)-value among all simple connected outerplanar (resp. planar) graphs are determined.     

Neighbor product distinguishing total colorings

A total-[k]-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}\) such that any two adjacent elements in \(V (G) \cup E(G)\) receive different colors. Let f(v) denote the product of the color of a vertex v and the colors of all edges incident to v. A total-[k]-neighbor product distinguishing-coloring of G is a total-[k]-coloring of G such that \(f(u)\ne f(v)\), where \(uv\in E(G)\). By \(\chi ^{\prime \prime }_{\prod }(G)\), we denote the smallest value k in such a coloring of G. We conjecture that \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that the conjecture holds for complete graphs, cycles, trees, bipartite graphs and subcubic graphs. Furthermore, we show that if G is a \(K_4\)-minor free graph with \(\Delta (G)\ge 4\), then \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2\).