Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Let \(\mathcal{C}\) be a uniform clutter and let A be the incidence matrix of \(\mathcal{C}\). We denote the column vectors of A by v ₁,…,v _q . Under certain conditions we prove that \(\mathcal{C}\) is vertex critical. If \(\mathcal{C}\) satisfies the max-flow min-cut property, we prove that A diagonalizes over ? to an identity matrix and that v ₁,…,v _q form a Hilbert basis. We also prove that if \(\mathcal{C}\) has a perfect matching such that \(\mathcal{C}\) has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ? to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v ₁,…,v _q is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.     

Semi-online scheduling on two uniform machines with the known largest size

This paper investigates semi-online scheduling on two uniform machines with the known largest size. Denote by s _j the speed of each machine, j=1,2. Assume 0<s ₁≤s ₂, and let s=s ₂/s ₁ be the speed ratio. First, for the speed ratio \(s\in [1,\sqrt{2}]\), we present an optimal semi-online algorithm \(\mathcal{LSMP}\) with the competitive ratio \(\mathrm{max}\{\frac {2(s+1)}{2s+1},s\}\). Second, we present a semi-online algorithm \(\mathcal{HSMP}\). And for \(s\in(\sqrt{2},1+\sqrt{3})\), the competitive ratio of \(\mathcal{HSMP}\) is strictly smaller than that of the online algorithm \(\mathcal{LS}\). Finally, for the speed ratio s≥s ^*≈3.715, we show that the known largest size cannot help us to design a semi-online algorithm with the competitive ratio strictly smaller than that of \(\mathcal{LS}\). Moreover, we show a lower bound for \(s\in(\sqrt{2},s^{*})\).     

An improved parameterized algorithm for the <Emphasis Type="Italic">p</Emphasis>-cluster vertex deletion problem

In the p-Cluster Vertex Deletion problem, we are given a graph \(G=(V,E)\) and two parameters k and p, and the goal is to determine if there exists a subset X of at most k vertices such that the removal of X results in a graph consisting of exactly p disjoint maximal cliques. Let \(r=p/k\). In this paper, we design a branching algorithm with time complexity \(O(\alpha ^k+|V||E|)\), where \(\alpha \) depends on r and has a rough upper bound \(\min \{1.618^{1+r},2\}\). With a more precise analysis, we show that \(\alpha =1.28\cdot 3.57^{r}\) for \(r\le 0.219\); \(\alpha =(1-r)^{r-1}r^{-r}\) for \(0.219< r<1/2\); and \(\alpha =2\) for \(r\ge 1/2\), respectively. Our algorithm also works with the same time complexity for the variant that the number of clusters is at most p. Our result improves the previous best time complexity \(O^*(1.84^{p+k})\) and implies that for fixed p the problem can be solved as efficiently as Vertex Cover.     

Cost sharing on prices for games on graphs

Let \(N=\{1,\dots ,n\}\) be a set of customers who want to buy a single homogenous goods in market. Let \(q_i>0\) be the quantity that \(i\in N\) demands, \(q=(q_1,\dots ,q_n)\) and \(q_S=\sum _{i\in S}q_i\) for \(S\subseteq N\). If f(s) is a (increasing and concave) cost function, then it yields a cooperative game (N, f, q) by defining characteristic function \(v(S)=f(q_S)\) for \(S\subseteq N\). We now consider the way of taking packages of goods by customers and define a communication graph L on N, in which i and j are linked if they can take packages for each other. So if i and j are connected, then a package can be delivered from i to j by some intermediators. We thus admit any connected subset as a feasible coalition, and obtain a game (N, f, q, L) by defining characteristic function \(v_L(S)=\sum _{R\in S/L}f(q_R)\) for \(S\subseteq N\), where S / L is the family of induced components (maximal connected subset) in S. It is shown that there is an allocation (cost shares) \(x=(x_1,\dots ,x_n)\) from the core for the game (\(x_S\le v_L(S)\) for any \(S\subseteq N\)) such that x satisfies Component Efficiency and Ranking for Unit Prices. If f(s) and q satisfy some further condition, then there is an allocation x from the core such that x satisfies Component Efficiency, and \(x_i \le x_j\) and \(\frac{x_i}{q_i} \ge \frac{x_j}{q_j}\) if \(q_i \le q_j\) for i and j in the same component of N.     

Nordhaus–Gaddum bounds for total Roman domination

A Nordhaus–Gaddum-type result is a lower or an upper bound on the sum or the product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total Roman domination number \(\gamma _{tR}\). Let G be a graph on n vertices and let \(\overline{G}\) denote the complement of G, and let \(\delta ^*(G)\) denote the minimum degree among all vertices in G and \(\overline{G}\). For \(\delta ^*(G)\ge 1\), we show that (i) if G and \(\overline{G}\) are connected, then \((\gamma _{tR}(G)-4)(\gamma _{tR}(\overline{G})-4)\le 4\delta ^*(G)-4\), (ii) if \(\gamma _{tR}(G), \gamma _{tR}(\overline{G})\ge 8\), then \(\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le 2\delta ^*(G)+5\) and (iii) \(\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le n+5\) and \(\gamma _{tR}(G)\gamma _{tR}(\overline{G})\le 6n-5\).     

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 \(\max \{b,4\}\).     

Separating NE from some nonuniform nondeterministic complexity classes

We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) \(\mathrm{NE}\not\subseteq R^{\mathrm{NP}}_{n^{o(1)}-T}(\mathrm{TALLY})\); (2) \(\mathrm{NE}\not\subseteq R^{SN}_{m}(\mathrm{SPARSE})\); (3) \(\mathrm{NEXP}\not\subseteq \mathrm{P}^{\mathrm{NP}}_{n^{k}-T}/n^{k}\) for all k≥1; and (4) \(\mathrm{NE}\not\subseteq \mathrm{P}_{btt}(\mathrm{NP}\oplus\mathrm{SPARSE})\). Result (3) extends a previous result by Mocas to nonuniform reductions. We also investigate how different an NE-hard set is from an NP-set. We show that for any NP subset A of a many-one-hard set H for NE, there exists another NP subset A′ of H such that A′? A and A′?A is not of sub-exponential density.     

Sparse multipartite graphs as partition universal for graphs with bounded degree

For graphs G and H, let \(G\rightarrow (H,H)\) signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote \(\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}\). For any \(\Delta \) and n, we say that G is partition universal for \(\mathcal {H}(\Delta ,n)\) if \(G\rightarrow (H,H)\) for every \(H\in \mathcal {H}(\Delta ,n)\). Let \(G_r(N,p)\) be the random spanning subgraph of the complete r-partite graph \(K_r(N)\) with N vertices in each part, in which each edge of \(K_r(N)\) appears with probability p independently and randomly. We prove that for fixed \(\Delta \ge 2\) there exist constants r, B and C depending only on \(\Delta \) such that if \(N\ge Bn\) and \(p=C(\log N/N)^{1/\Delta }\), then asymptotically almost surely \(G_r(N,p)\) is partition universal for \(\mathcal {H}(\Delta ,n)\).     

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.     

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.     

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 b-coloring of tight graphs

A coloring c of a graph \(G=(V,E)\) is a b -coloring if for every color i there is a vertex, say w(i), of color i whose neighborhood intersects every other color class. The vertex w(i) is called a b-dominating vertex of color i. The b -chromatic number of a graph G, denoted by b(G), is the largest integer k such that G admits a b-coloring with k colors. Let m(G) be the largest integer m such that G has at least m vertices of degree at least \(m-1\). A graph G is tight if it has exactly m(G) vertices of degree \(m(G)-1\), and any other vertex has degree at most \(m(G)-2\). In this paper, we show that the b-chromatic number of tight graphs with girth at least 8 is at least \(m(G)-1\) and characterize the graphs G such that \(b(G)=m(G)\). Lin and Chang (2013) conjectured that the b-chromatic number of any graph in \(\mathcal {B}_{m}\) is m or \(m-1\) where \(\mathcal {B}_{m}\) is the class of tight bipartite graphs \((D,D{^\prime })\) of girth 6 such that D is the set of vertices of degree \(m-1\). We verify the conjecture of Lin and Chang for some subclass of \(\mathcal {B}_{m}\), and we give a lower bound for any graph in \(\mathcal {B}_{m}\).     

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.     

Minimum 2-distance coloring of planar graphs and channel assignment

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree \(\Delta \), \(\chi _2(G) \le 7\) if \(\Delta \le 3\), \(\chi _2(G) \le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\). In this paper, we prove that: (1) If G is a planar graph with maximum degree \(\Delta \le 5\), then \(\chi _{2}(G)\le 20\); (2) If G is a planar graph with maximum degree \(\Delta \ge 6\), then \(\chi _{2}(G)\le 5\Delta -7\).     

Lambda number for the direct product of some family of graphs

An L(2, 1)-labeling for a graph \(G=(V,E)\) is a function f on V such that \(|f(u)-f(v)|\ge 2\) if u and v are adjacent and f(u) and f(v) are distinct if u and v are vertices of distance two. The L(2, 1)-labeling number, or the lambda number \(\lambda (G)\), for G is the minimum span over all L(2, 1)-labelings of G. When \(P_{m}\times C_{n}\) is the direct product of a path \(P_m\) and a cycle \(C_n\), Jha et al. (Discret Appl Math 145:317–325, 2005) computed the lambda number of \(P_{m}\times C_{n}\) for \(n\ge 3\) and \(m=4,5\). They also showed that when \(m\ge 6\) and \(n\ge 7\), \(\lambda (P_{m}\times C_{n})=6\) if and only if n is the multiple of 7 and conjectured that it is 7 if otherwise. They also showed that \(\lambda (C_{7i}\times C_{7j})=6\) for some i, j. In this paper, we show that when \(m\ge 6\) and \(n\ge 3\), \(\lambda (P_m\times C_n)=7\) if and only if n is not a multiple of 7. Consequently the conjecture is proved. Here we also provide the conditions on m and n such that \(\lambda (C_m\times C_n)\le 7\).     

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s ₁,…,s _d }?V and a function f : S→? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ∑ _i=1 ^d f(s _i ) in-trees denoted by \(T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}\) for every i=1,…,d such that \(T_{i,1},\ldots,T_{i,f(s_{i})}\) are rooted at s _i , each T _i,j spans vertices from which s _i is reachable and the union of all arc sets of T _i,j for i=1,…,d and j=1,…,f(s _i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑ _i=1 ^d f(s _i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.     

On zero-sum $$\mathbb {Z}_{2j}^k$$-magic graphs

Let \(G = (V,E)\) be a finite graph and let \((\mathbb {A},+)\) be an abelian group with identity 0. Then G is \(\mathbb {A}\)-magic if and only if there exists a function \(\phi \) from E into \(\mathbb {A} - \{0\}\) such that for some \(c \in \mathbb {A}, \sum _{e \in E(v)} \phi (e) = c\) for every \(v \in V\), where E(v) is the set of edges incident to v. Additionally, G is zero-sum \(\mathbb {A}\)-magic if and only if \(\phi \) exists such that \(c = 0\). We consider zero-sum \(\mathbb {A}\)-magic labelings of graphs, with particular attention given to \(\mathbb {A} = \mathbb {Z}_{2j}^k\). For \(j \ge 1\), let \(\zeta _{2j}(G)\) be the smallest positive integer c such that G is zero-sum \(\mathbb {Z}_{2j}^c\)-magic if c exists; infinity otherwise. We establish upper bounds on \(\zeta _{2j}(G)\) when \(\zeta _{2j}(G)\) is finite, and show that \(\zeta _{2j}(G)\) is finite for all r-regular \(G, r \ge 2\). Appealing to classical results on the factors of cubic graphs, we prove that \(\zeta _4(G) \le 2\) for a cubic graph G, with equality if and only if G has no 1-factor. We discuss the problem of classifying cubic graphs according to the collection of finite abelian groups for which they are zero-sum group-magic.     

Majorization and the spectral radius of starlike trees

A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by \(\lambda (G)\), is the largest eigenvalue of G. Let k and \(n_1,\ldots ,n_k\) be some positive integers. Let \(T(n_1,\ldots ,n_k)\) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1-1},\ldots ,P_{n_k-1}\) where every neighbor of v in T has degree one or two. Let \(P=(p_1,\ldots ,p_k)\) and \(Q=(q_1,\ldots ,q_k)\), where \(p_1\ge \cdots \ge p_k\ge 1\) and \(q_1\ge \cdots \ge q_k\ge 1\) are integer. We say P majorizes Q and let \(P\succeq _M Q\), if for every j, \(1\le j\le k\), \(\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i\), with equality if \(j=k\). In this paper we show that if P majorizes Q, that is \((p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)\), then \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).     

Neighbor-sum-distinguishing edge choosability of subcubic graphs

A graph G is said to be neighbor-sum-distinguishing edge k-choose if, for every list L of colors such that L(e) is a set of k positive real numbers for every edge e, there exists a proper edge coloring which assigns to each edge a color from its list so that for each pair of adjacent vertices u and v the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\) denote the smallest integer k such that G is neighbor-sum-distinguishing edge k-choose. In this paper, we prove that if G is a subcubic graph with the maximum average degree mad(G), then (1) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 7\); (2) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 6\) if \(\hbox {mad}(G)<\frac{36}{13}\); (3) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 5\) if \(\hbox {mad}(G)<\frac{5}{2}\).     

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 \(F\cong K_{k+1,k+2}\).