Neighbor sum distinguishing total choosability of planar graphs

A total-k-coloring of a graph G is a mapping \(c: V(G)\cup E(G)\rightarrow \{1, 2,\dots , k\}\) such that any two adjacent or incident elements in \(V(G)\cup E(G)\) receive different colors. For a total-k-coloring of G, let \(\sum _c(v)\) denote the total sum of colors of the edges incident with v and the color of v. If for each edge \(uv\in E(G)\), \(\sum _c(u)\ne \sum _c(v)\), then we call such a total-k-coloring neighbor sum distinguishing. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by \(\chi _{\Sigma }^{''}(G)\). Pil?niak and Wo?niak conjectured \(\chi _{\Sigma }^{''}(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that for any planar graph G with maximum degree \(\Delta (G)\), \(ch^{''}_{\Sigma }(G)\le \max \{\Delta (G)+3,16\}\), where \(ch^{''}_{\Sigma }(G)\) is the neighbor sum distinguishing total choosability of G.     

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 \(G\).     

The minimum vulnerability problem on specific graph classes

Suppose that each edge e of an undirected graph G is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of e, respectively. Then, we consider the problem of finding \(k\) paths in G between two prescribed vertices with the minimum total cost; each edge e can be shared without any cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost for e. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.     

An inequality that relates the size of a bipartite graph with its order and restrained domination number

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 \(G\). Consider a bipartite graph \(G\) of order \(n\ge 4,\) and let \(k\in \{2,3,...,n-2\}.\) In this paper we will show that if \(\gamma _{r}(G)=k\), then \(m\le ((n-k)(n-k+6)+4k-8)/4\). We will also show that this bound is best possible.     

Equitable colorings of Cartesian products of square of cycles and paths with complete bipartite graphs

A graph G is said to be equitably k-colorable if the vertex set of G can be divided into k independent sets for which any two sets differ in size at most one. The equitable chromatic number of G, \(\chi _{=}(G)\), is the minimum k for which G is equitably k-colorable. The equitable chromatic threshold of G, \(\chi _{=}^{*}(G)\), is the minimum k for which G is equitably \(k'\)-colorable for all \(k'\ge k\). In this paper, the exact values of \(\chi _{=}^{*}(G\Box H)\) and \(\chi _{=}(G\Box H)\) are obtained when G is the square of a cycle or a path and H is a complete bipartite graph.     

L(2,1)-labelings of the edge-multiplicity-paths-replacement of a graph

An L(2,1)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(2\), and the difference between labels of vertices that are distance two apart is at least 1. The span of an L(2,1)-labeling of a graph \(G\) is the difference between the maximum and minimum integers used by it. The minimum span of an L(2,1)-labeling of \(G\) is denoted by \(\lambda (G)\). This paper focuses on L(2,1)-labelings-number of the edge-multiplicity-paths-replacement \(G(rP_{k})\) of a graph \(G\). In this paper, we obtain that \( r\Delta +1 \le \lambda (G(rP_{5}))\le r\Delta +2\), \(\lambda (G(rP_{k}))= r\Delta +1\) for \(k\ge 6\); and \(\lambda (G(rP_{4}))\le (\Delta +1)r+1\), \(\lambda (G(rP_{3}))\le (\Delta +1)r+\Delta \) for any graph \(G\) with maximum degree \(\Delta \). And the L(2,1)-labelings-numbers of the edge-multiplicity-paths-replacement \(G(rP_{k})\) are completely determined for \(1\le \Delta \le 2\). And we show that the class of graphs \(G(rP_{k})\) with \(k\ge 3 \) satisfies the conjecture: \(\lambda ^{T}_{2}(G)\le \Delta +2\) by Havet and Yu (Technical Report 4650, 2002).     

Atoms of cyclic edge connectivity in regular graphs

A cyclic edge-cut of a connected graph \(G\) is an edge set, the removal of which separates two cycles. If \(G\) has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph \(G\), the cyclic edge connectivity of a graph \(G\), denoted by \(\lambda _c(G)\), is the minimum cardinality over all cyclic edge cuts. Let \(X\) be a non-empty proper subset of \(V(G)\). If \([X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}\) is a minimum cyclic edge cut of \(G\), then \(X\) is called a \(\lambda _c\) -fragment of \(G\). A \(\lambda _c\)-fragment with minimum cardinality is called a \(\lambda _c\) -atom. Let \(G\) be a \(k (k\ge 3)\)-regular cyclically separable graph with \(\lambda _c(G)<g(k-2)\), where \(g\) is the girth of \(G\). A combination of the results of Nedela and Skoviera (Math Slovaca 45:481–499, 1995) and Xu and Liu (Australas J Combin 30:41–49, 2004) gives that if \(k\ne 5\) then any two distinct \(\lambda _c\)-atoms of \(G\) are disjoint. The remaining case of \(k=5\) is considered in this paper, and a new proof for Nedela and ?koviera’s result is also given. As a result, we obtain the following result. If \(X\) and \(X'\) are two distinct \(\lambda _c\)-atoms of \(G\) such that \(X\cap X'\ne \emptyset \), then \((k,g)=(5,3)\) and \(G[X]\cong K_4\). As corollaries, several previous results are easily obtained.     

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.     

Further properties on the degree distance of graphs

In this paper, we study the degree distance of a connected graph \(G\), defined as \(D^{'} (G)=\sum _{u\in V(G)} d_{G} (u)D_{G} (u)\), where \(D_{G} (u)\) is the sum of distances between the vertex \(u\) and all other vertices in \(G\) and \(d_{G} (u)\) denotes the degree of vertex \(u\) in \(G\). Our main purpose is to investigate some properties of degree distance. We first investigate degree distance of tensor product \(G\times K_{m_0,m_1,\cdots ,m_{r-1}}\), where \(K_{m_0,m_1,\cdots ,m_{r-1}}\) is the complete multipartite graph with partite sets of sizes \(m_0,m_1,\cdots ,m_{r-1}\), and we present explicit formulas for degree distance of the product graph. In addition, we give some Nordhaus–Gaddum type bounds for degree distance. Finally, we compare the degree distance and eccentric distance sum for some graph families.     

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.     

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 \(m(G)=q(n,k)\).     

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.     

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.     

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.     

On the efficiency index of a graph

A graph \(G\) has an efficient dominating set \(D \subseteq V(G)\) if \(D\) dominates every vertex exactly once. In this paper we introduce the study of the family \({S_k}\) of graphs for which every \(G-S\) is efficiently dominatable for \(0 \le |S|\le k\). Assuming that \(G\) is efficiently dominatable, the efficiency index is the largest value k for which \(G\) is in \(S_k\). A graph \(G\) will be called super-efficient if every induced subgraph is efficiently dominatable. We give some characterizations for trees, grids, cylinders and torii to be super-efficient.     

2-Distance coloring of a planar graph without 3, 4, 7-cycles

A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most two receive distinct colors. The 2-distance chromatic number \(\chi _{2}(G)\) is the smallest k such that G is k-2-distance colorable. In this paper, we prove that every planar graph without 3, 4, 7-cycles and \(\Delta (G)\ge 15\) is (\(\Delta (G)+4\))-2-distance colorable.     

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 $$(3,1)^{*}$$-choosable theorem on planar graphs

A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex \(v\in V(G)\). An \((L,d)^*\)-coloring is a mapping \(\pi \) that assigns a color \(\pi (v)\in L(v)\) to each vertex \(v\in V(G)\) so that at most d neighbors of v receive color \(\pi (v)\). A graph G is said to be \((k,d)^*\)-choosable if it admits an \((L,d)^*\)-coloring for every list assignment L with \(|L(v)|\ge k\) for all \(v\in V(G)\). In this paper, we prove that every planar graph with neither adjacent triangles nor 6-cycles is \((3,1)^*\)-choosable. This is a partial answer to a question of Xu and Zhang (Discret Appl Math 155:74–78, 2007) that every planar graph without adjacent triangles is \((3,1)^*\)-choosable. Also, this improves a result in Lih et al. (Appl Math Lett 14:269–273, 2001) which says that every planar graph without 4- and 6-cycles is \((3,1)^*\)-choosable.     

An $$O^{*}(1.4366^n)$$-time exact algorithm for maximum $$P_2$$P2-packing in cubic graphs

Given a graph \(G=(V, E)\), a \(P_2\)-packing \(\mathcal {P}\) is a collection of vertex disjoint copies of \(P_2\)s in \(G\) where a \(P_2\) is a simple path with three vertices and two edges. The Maximum \(P_2\)-Packing problem is to find a \(P_2\)-packing \(\mathcal {P}\) in the input graph \(G\) of maximum cardinality. This problem is NP-hard for cubic graphs. In this paper, we give a branch-and-reduce algorithm for the Maximum \(P_2\)-Packing problem in cubic graphs. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time \(O^{*}(1.4366^n)\) which is faster than previous known exact algorithms where \(n\) is the number of vertices in the input graph.     

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.