Modified linear programming and class 0 bounds for graph pebbling

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move removes two pebbles from some vertex and places one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert’s paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every n-vertex Class 0 graph has at least \(\frac{5}{3}n - \frac{11}{3}\) edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to \(2n - 5\), which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.     

On the odd girth and the circular chromatic number of generalized Petersen graphs

A class \(\mathcal{G}\) of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph \(\mathrm {G} \in \mathcal{G}\) such that the girth (odd-girth) of \(\mathrm {G}\) is \(\ge g\). A girth-closed (odd-girth-closed) class \(\mathcal{G}\) of graphs is said to be pentagonal (odd-pentagonal) if there exists a positive integer \(g^*\) depending on \(\mathcal{G}\) such that any graph \(\mathrm {G} \in \mathcal{G}\) whose girth (odd-girth) is greater than \(g^*\) admits a homomorphism to the five cycle (i.e. is \(\mathrm {C}_{_{5}}\)-colourable). Although, the question “Is the class of simple 3-regular graphs pentagonal?” proposed by Ne?et?il (Taiwan J Math 3:381–423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has shown that there exists an odd-girth-closed subclass of simple 3-regular graphs which is not odd-pentagonal. In this article, motivated by the conjecture that the class of generalized Petersen graphs is odd-pentagonal, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using the combinatorial and number theoretic properties of this problem, we explicitly compute the odd girth of such graphs, showing that the class is odd-girth-closed. Also, we obtain upper and lower bounds for the circular chromatic number of these graphs, and as a consequence, we show that the subclass containing generalized Petersen graphs \(\mathrm {Pet}(n,k)\) for which either k is even, n is odd and \(n\mathop {\equiv }\limits ^{k-1}\pm 2\) or both n and k are odd and \(n\ge 5k\) is odd-pentagonal. This in particular shows the existence of nontrivial odd-pentagonal subclasses of 3-regular simple graphs.     

The <Emphasis Type="Italic">k</Emphasis>-hop connected dominating set problem: approximation and hardness

Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (\({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\)). We prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {NP}\)-hard on planar bipartite graphs of maximum degree 4. We also prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {APX}\)-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) on bipartite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complexity of computing this graph parameter. On the positive side, we show an approximation algorithm for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\). Finally, when \(k=1\), we present two new approximation algorithms for the weighted version of the problem restricted to graphs with a polynomially bounded number of minimal separators.     

Fractional matching preclusion of graphs

Let F be an edge subset and \(F^{\prime }\) a subset of edges and vertices of a graph G. If \(G-F\) and \(G-F^{\prime }\) have no fractional perfect matchings, then F is a fractional matching preclusion (FMP) set and \(F^{\prime }\) is a fractional strong MP (FSMP) set of G. The FMP (FSMP) number of G is the minimum size of FMP (FSMP) sets of G. In this paper, the FMP number and the FSMP number of Petersen graph, complete graphs and twisted cubes are obtained, respectively.     

A note on optimal pebbling of hypercubes

A pebbling move consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. If a distribution δ of pebbles lets us move at least one pebble to each vertex by applying pebbling moves repeatedly(if necessary), then δ is called a pebbling of G. The optimal pebbling number f′(G) of G is the minimum number of pebbles used in a pebbling of G. In this paper, we improve the known upper bound for the optimal pebbling number of the hypercubes Q _n . Mainly, we prove for large n, $f'(Q_{n})=O(n^{3/2}(\frac {4}{3})^{n})$ by a probabilistic argument.     

Coupon coloring of some special graphs

Let G be a graph without isolated vertices. A k-coupon coloring of G is a k-coloring of G such that the neighborhood of every vertex of G contains vertices of all colors from \([k] =\{1, 2, \ldots , k\}\), which was recently introduced by Chen, Kim, Tait and Verstraete. The coupon coloring number \(\chi _c(G)\) of G is the maximum k for which a k-coupon coloring exists. In this paper, we mainly study the coupon coloring of some special classes of graphs. We determine the coupon coloring numbers of complete graphs, complete k-partite graphs, wheels, cycles, unicyclic graphs, bicyclic graphs and generalised \(\Theta \)-graphs.     

On maximum Wiener index of trees and graphs with given radius

Let G be a connected graph of order n. The long-standing open and close problems in distance graph theory are: what is the Wiener index W(G) or average distance \(\mu (G)\) among all graphs of order n with diameter d (radius r)? There are very few number of articles where were worked on the relationship between radius or diameter and Wiener index. In this paper, we give an upper bound on Wiener index of trees and graphs in terms of number of vertices n, radius r, and characterize the extremal graphs. Moreover, from this result we give an upper bound on \(\mu (G)\) in terms of order and independence number of graph G. Also we present another upper bound on Wiener index of graphs in terms of number of vertices n, radius r and maximum degree \(\Delta \), and characterize the extremal graphs.     

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 \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2\).     

Embeddings into almost self-centered graphs of given radius

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius r is called an r-ASC graph. The r-ASC index \(\theta _r(G)\) of a graph G is the minimum number of vertices needed to be added to G such that an r-ASC graph is obtained that contains G as an induced subgraph. It is proved that \(\theta _r(G)\le 2r\) holds for any graph G and any \(r\ge 2\) which improves the earlier known bound \(\theta _r(G)\le 2r+1\). It is further proved that \(\theta _r(G)\le 2r-1\) holds if \(r\ge 3\) and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that \(\theta _3(G)\in \{3,4\}\) if G has diameter 2 and for several classes of graphs of diameter 2 the exact value of the 3-ASC index is obtained. For instance, if a graph G of diameter 2 does not contain a diametrical triple, then \(\theta _3(G) = 4\). The 3-ASC index of paths of order \(n\ge 1\), cycles of order \(n\ge 3\), and trees of order \(n\ge 10\) and diameter \(n-2\) are also determined, respectively, and several open problems proposed.     

On the coefficients of the independence polynomial of graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let \(i_k = i_k(G)\) be the number of independent sets of cardinality k of G. The independence polynomial \(I(G, x)=\sum _{k\geqslant 0}i_k(G)x^k\) defined first by Gutman and Harary has been the focus of considerable research recently, whereas \(i(G)=I(G, 1)\) is called the Merrifield–Simmons index of G. In this paper, we first proved that among all trees of order n,  the kth coefficient \(i_k\) is smallest when the tree is a path, and is largest for star. Moreover, the graph among all trees of order n with diameter at least d whose all coefficients of I(G, x) are largest is identified. Then we identify the graphs among the n-vertex unicyclic graphs (resp. n-vertex connected graphs with clique number \(\omega \)) which simultaneously minimize all coefficients of I(G, x), whereas the opposite problems of simultaneously maximizing all coefficients of I(G, x) among these two classes of graphs are also solved respectively. At last we characterize the graph among all the n-vertex connected graph with chromatic number \(\chi \) (resp. vertex connectivity \(\kappa \)) which simultaneously minimize all coefficients of I(G, x). Our results may deduce some known results on Merrifield–Simmons index of graphs.     

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(\overline{3K_2}\)-free co-bipartite graphs and polynomially solvable on \(C_4\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most \(2^k\) vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most \(2^k\) vertices. Finally we determine all graphs of small complete width \(k\le 3\).     

First-Fit colorings of graphs with no cycles of a prescribed even length

The First-Fit (or Grundy) chromatic number of a graph G denoted by \(\chi _{{_\mathsf{FF}}}(G)\), is the maximum number of colors used by the First-Fit (greedy) coloring algorithm when applied to G. In this paper we first show that any graph G contains a bipartite subgraph of Grundy number \(\lfloor \chi _{{_\mathsf{FF}}}(G) /2 \rfloor +1\). Using this result we prove that for every \(t\ge 2\) there exists a real number \(c>0\) such that in every graph G on n vertices and without cycles of length 2t, any First-Fit coloring of G uses at most \(cn^{1/t}\) colors. It is noted that for \(t=2\) this bound is the best possible. A compactness conjecture is also proposed concerning the First-Fit chromatic number involving the even girth of graphs.     

An $$O(|E(G)|^2)$$ algorithm for recognizing Pfaffian graphs of a type of bipartite graphs

A graph \(G=(V,E)\) with even number vertices is called Pfaffian if it has a Pfaffian orientation, namely it admits an orientation such that the number of edges of any M-alternating cycle which have the same direction as the traversal direction is odd for some perfect matching M of the graph G. In this paper, we obtain a necessary and sufficient condition of Pfaffian graphs in a type of bipartite graphs. Then, we design an \(O(|E(G)|^2)\) algorithm for recognizing Pfaffian graphs in this class and constructs a Pfaffian orientation if the graph is Pfaffian. The results improve and generalize some known results.     

Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles

A total coloring of a graph G is an assignment of colors to the vertices and the edges of G such that every pair of adjacent/incident elements receive distinct colors. The total chromatic number of a graph G, denoted by \(\chi ''(G)\), is the minimum number of colors in a total coloring of G. The well-known total coloring conjecture (TCC) says that every graph with maximum degree \(\Delta \) admits a total coloring with at most \(\Delta + 2\) colors. A graph is 1-toroidal if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 1-toroidal graphs, and prove that the TCC holds for the 1-toroidal graphs with maximum degree at least 11 and some restrictions on the triangles. Consequently, if G is a 1-toroidal graph with maximum degree \(\Delta \) at least 11 and without adjacent triangles, then G admits a total coloring with at most \(\Delta + 2\) colors.     

On the algorithmic aspects of strong subcoloring

A partition of the vertex set V(G) of a graph G into \(V(G)=V_1\cup V_2\cup \cdots \cup V_k\) is called a k-strong subcoloring if \(d(x,y)\ne 2\) in G for every \(x,y\in V_i\), \(1\le i \le k\) where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as \(\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k\)-\(\text {strong subcoloring}\}\). In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of \(K_{1,r}\)-free split graphs, StrongSubcoloring is in P when \(r\le 3\) and NP-complete when \(r>3\) and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of \(P_4\); otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every \(x,y\in S\), \(d(x,y)= 2\) in G and the cardinality of a maximum strong set in G is denoted by \(\alpha _{s}(G)\). Clearly, \(\alpha _{s}(G)\le \chi _{sc}(G)\). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) \(K_{1,4}\)-free split graphs, and it is polynomial time solvable for (i) trees and (ii) \(P_4\)-free graphs.     

Cha n nel assig nme n t problem a nd n-fold t-separa ted $$L(j_1,j_2,\ldo ts,j_m)$$-labeli ng of graphs

This paper considers the channel assignment problem in mobile communications systems. Suppose there are many base stations in an area, each of which demands a number of channels to transmit signals. The channels assigned to the same base station must be separated in some extension, and two channels assigned to two different stations that are within a distance must be separated in some other extension according to the distance between the two stations. The aim is to assign channels to stations so that the interference is controlled within an acceptable level and the spectrum of channels used is minimized. This channel assignment problem can be modeled as the multiple t-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling of the interference graph. In this paper, we consider the case when all base stations demand the same number of channels. This case is referred as n-fold t-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling of a graph. This paper first investigates the basic properties of n-fold t-separated \(L(j_1,j_2,\ldots ,j_m)\)-labelings of graphs. And then it focuses on the special case when \(m=1\). The optimal n-fold t-separated L(j)-labelings of all complete graphs and almost all cycles are constructed. As a consequence, the optimal n-fold t-separated \(L(j_1,j_2,\ldots ,j_m)\)-labelings of the triangular lattice and the square lattice are obtained for the case \(j_1=j_2=\cdots =j_m\). This provides an optimal solution to the corresponding channel assignment problems with interference graphs being the triangular lattice and the square lattice, in which each base station demands a set of n channels that are t-separated and channels from two different stations at distance at most m must be \(j_1\)-separated. We also study a variation of n-fold t-separated \(L(j_1,j_2,\ldots ,j_m)\)-labeling, namely, n-fold t-separated consecutive \(L(j_1,j_2,\ldots ,j_m)\)-labeling. And present the optimal n-fold t-separated consecutive L(j)-labelings of all complete graphs and cycles.     

Is there any polynomial upper bound for the universal labeling of graphs?

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation \(\ell \) of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from \(\{1,2,\ldots , k\}\) denoted it by \(\overrightarrow{\chi _{u}}(G) \). We have \(2\Delta (G)-2 \le \overrightarrow{\chi _{u}} (G)\le 2^{\Delta (G)}\), where \(\Delta (G)\) denotes the maximum degree of G. In this work, we offer a provocative question that is: “Is there any polynomial function f such that for every graph G, \(\overrightarrow{\chi _{u}} (G)\le f(\Delta (G))\)?”. Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree T, \(\overrightarrow{\chi _{u}}(T)={\mathcal {O}}(\Delta ^3) \). Next, we show that for a given 3-regular graph G, the universal labeling number of G is 4 if and only if G belongs to Class 1. Therefore, for a given 3-regular graph G, it is an \( {{\mathbf {N}}}{{\mathbf {P}}} \)-complete to determine whether the universal labeling number of G is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.     

Safe sets in graphs: Graph classes and structural parameters

A safe set of a graph \(G=(V,E)\) is a non-empty subset S of V such that for every component A of G[S] and every component B of \(G[V {\setminus } S]\), we have \(|A| \ge |B|\) whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between the tree-depth and the vertex cover number. We then conclude the paper by showing hardness for split graphs and planar graphs.