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 \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).     

The longest commonly positioned increasing subsequences problem

Based on the well-known longest increasing subsequence problem and longest common increasing subsequence (LCIS) problem, we propose the longest commonly positioned increasing subsequences (LCPIS) problem. Let \(A=\langle a_1,a_2,\ldots ,a_n\rangle \) and \(B{=}\left\langle b_1,b_2,\ldots ,b_n\right\rangle \) be two input sequences. Let \({ Asub}=\left\langle a_{i_1},a_{i_2},\ldots ,a_{i_l}\right\rangle \) be a subsequence of A and \({ Bsub}=\left\langle b_{j_1},b_{j_2},\ldots ,b_{j_l}\right\rangle \) be a subsequence of B such that \(a_{i_k}\le a_{i_{k+1}}, b_{j_k}\le b_{j_{k+1}}(1\le k<l)\), and \(a_{i_k}\) and \(b_{j_k}\) (\(1\le k\le l\)) are commonly positioned (have the same index \(i_k=j_k\)) in A and B respectively but these two elements do not need to be equal. The LCPIS problem aims at finding a pair of subsequences Asub and \({ Bsub}\) as long as possible. When all the elements of the two input sequences are positive integers, this paper presents an algorithm with \(O(n\log n \log \log M)\) time to compute the LCPIS, where \(M={ min}\{{ max}_{1\le i\le n}a_i,{ max}_{1\le j\le n}b_j\}\). And we also show a dual relationship between the LCPIS problem and the LCIS problem.     

Planar graphs without 4-cycles and close triangles are (2, 0, 0)-colorable

For a set of nonnegative integers \(c_1, \ldots , c_k\), a \((c_1, c_2,\ldots , c_k)\)-coloring of a graph G is a partition of V(G) into \(V_1, \ldots , V_k\) such that for every i, \(1\le i\le k, G[V_i]\) has maximum degree at most \(c_i\). We prove that all planar graphs without 4-cycles and no less than two edges between triangles are (2, 0, 0)-colorable.     

On the <Emphasis Type="Italic">L</Emphasis>(2, 1)-labeling conjecture for brick product graphs

Let \(G=(V, E)\) be a graph. Denote \(d_G(u, v)\) the distance between two vertices u and v in G. An L(2, 1)-labeling of G is a function \(f: V \rightarrow \{0,1,\ldots \}\) such that for any two vertices u and v, \(|f(u)-f(v)| \ge 2\) if \(d_G(u, v) = 1\) and \(|f(u)-f(v)| \ge 1\) if \(d_G(u, v) = 2\). The span of f is the difference between the largest and the smallest number in f(V). The \(\lambda \)-number \(\lambda (G)\) of G is the minimum span over all L(2, 1)-labelings of G. In this paper, we conclude that the \(\lambda \)-number of each brick product graph is 5 or 6, which confirms Conjecture 6.1 stated in Li et al. (J Comb Optim 25:716–736, 2013).     

The $$(k,\ell )$$-proper index of graphs

A tree T in an edge-colored graph is called a proper tree if no two adjacent edges of T receive the same color. Let G be a connected graph of order n and k be an integer with \(2\le k \le n\). For \(S\subseteq V(G)\) and \(|S| \ge 2\), an S-tree is a tree containing the vertices of S in G. A set \(\{T_1,T_2,\ldots ,T_\ell \}\) of S-trees is called internally disjoint if \(E(T_i)\cap E(T_j)=\emptyset \) and \(V(T_i)\cap V(T_j)=S\) for \(1\le i\ne j\le \ell \). For a set S of k vertices of G, the maximum number of internally disjoint S-trees in G is denoted by \(\kappa (S)\). The k-connectivity \(\kappa _k(G)\) of G is defined by \(\kappa _k(G)=\min \{\kappa (S)\mid S\) is a k-subset of \(V(G)\}\). For a connected graph G of order n and for two integers k and \(\ell \) with \(2\le k\le n\) and \(1\le \ell \le \kappa _k(G)\), the \((k,\ell )\)-proper index \(px_{k,\ell }(G)\) of G is the minimum number of colors that are required in an edge-coloring of G such that for every k-subset S of V(G), there exist \(\ell \) internally disjoint proper S-trees connecting them. In this paper, we show that for every pair of positive integers k and \(\ell \) with \(k \ge 3\) and \(\ell \le \kappa _k(K_{n,n})\), there exists a positive integer \(N_1=N_1(k,\ell )\) such that \(px_{k,\ell }(K_n) = 2\) for every integer \(n \ge N_1\), and there exists also a positive integer \(N_2=N_2(k,\ell )\) such that \(px_{k,\ell }(K_{m,n}) = 2\) for every integer \(n \ge N_2\) and \(m=O(n^r) (r \ge 1)\). In addition, we show that for every \(p \ge c\root k \of {\frac{\log _a n}{n}}\) (\(c \ge 5\)), \(px_{k,\ell }(G_{n,p})\le 2\) holds almost surely, where \(G_{n,p}\) is the Erd?s–Rényi random graph model.     

On minimally 2-connected graphs with generalized connectivity $$\kappa _{3}=2$$

For \(S\subseteq G\), let \(\kappa (S)\) denote the maximum number r of edge-disjoint trees \(T_1, T_2, \ldots , T_r\) in G such that \(V(T_i)\cap V(T_j)=S\) for any \(i,j\in \{1,2,\ldots ,r\}\) and \(i\ne j\). For every \(2\le k\le n\), the k-connectivity of G, denoted by \(\kappa _k(G)\), is defined as \(\kappa _k(G)=\hbox {min}\{\kappa (S)| S\subseteq V(G)\ and\ |S|=k\}\). Clearly, \(\kappa _2(G)\) corresponds to the traditional connectivity of G. In this paper, we focus on the structure of minimally 2-connected graphs with \(\kappa _{3}=2\). Denote by \(\mathcal {H}\) the set of minimally 2-connected graphs with \(\kappa _{3}=2\). Let \(\mathcal {B}\subseteq \mathcal {H}\) and every graph in \(\mathcal {B}\) is either \(K_{2,3}\) or the graph obtained by subdividing each edge of a triangle-free 3-connected graph. We obtain that \(H\in \mathcal {H}\) if and only if \(H\in \mathcal {B}\) or H can be constructed from one or some graphs \(H_{1},\ldots ,H_{k}\) in \(\mathcal {B}\) (\(k\ge 1\)) by applying some operations recursively.     

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 total coloring of 2-degenerate graphs

A proper k-total coloring of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\ldots ,k\}\) such that no two adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let f(v) denote the sum of the colors on the edges incident with v and the color on vertex v. A proper k-total coloring of G is called neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). Let \(\chi ''_{\Sigma }(G)\) denote the smallest integer k in such a coloring of G. Pil?niak and Wo?niak conjectured that for any graph G, \(\chi ''_{\Sigma }(G)\le \Delta (G)+3\). In this paper, we show that if G is a 2-degenerate graph, then \(\chi ''_{\Sigma }(G)\le \Delta (G)+3\); Moreover, if \(\Delta (G)\ge 5\) then \(\chi ''_{\Sigma }(G)\le \Delta (G)+2\).     

Maximum coverage problem with group budget constraints

We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X, a collection \(\psi \) of subsets of X each of which is associated with a combinatorial structure such that for every set \(S_j\in \psi \), a cost \(c(S_j)\) can be calculated based on the combinatorial structure associated with \(S_j\), a partition \(G_1,G_2,\ldots ,G_l\) of \(\psi \), and budgets \(B_1,B_2,\ldots ,B_l\), and B. A solution to the problem consists of a subset H of \(\psi \) such that \(\sum _{S_j\in H} c(S_j) \le B\) and for each \(i \in {1,2,\ldots ,l}\), \(\sum _{S_j \in H\cap G_i}c(S_j)\le B_i\). The objective is to maximize \(|\bigcup _{S_j\in H}S_j|\). In our work we use a new and improved analysis of the greedy algorithm to prove that it is a \((\frac{\alpha }{3+2\alpha })\)-approximation algorithm, where \(\alpha \) is the approximation ratio of a given oracle which takes as an input a subset \(X^{new}\subseteq X\) and a group \(G_i\) and returns a set \(S_j\in G_i\) which approximates the optimal solution for \(\max _{D\in G_i}\frac{|D\cap X^{new}|}{c(D)}\). This analysis that is shown here to be tight for the greedy algorithm, improves by a factor larger than 2 the analysis of the best known approximation algorithm for MCG.     

Neighbor sum distinguishing total coloring of planar graphs without 4-cycles

Let \(G=(V,E)\) be a graph and \(\phi : V\cup E\rightarrow \{1,2,\ldots ,k\}\) be a proper total coloring of G. Let f(v) denote the sum of the color on a vertex v and the colors on all the edges incident with v. The coloring \(\phi \) is neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number of G, denoted by \(\chi _{\Sigma }''(G)\). Pil?niak and Wo?niak conjectured that \(\chi _{\Sigma }''(G)\le \Delta (G)+3\) for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that \(\chi _{\Sigma }''(G)\le \max \{\Delta (G)+2, 10\}\) for planar graph G without 4-cycles. The bound \(\Delta (G)+2\) is sharp if \(\Delta (G)\ge 8\).     

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.     

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.     

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.     

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 \(\lambda (C_m\times C_n)\le 7\).     

Bounds on the domination number of a digraph

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S is adjacent from at least one vertex in S. The domination number of D, denoted by \(\gamma (D)\), is the minimum cardinality of a dominating set of D. The Slater number \(s\ell (D)\) is the smallest integer t such that t added to the sum of the first t terms of the non-increasing out-degree sequence of D is at least as large as the order of D. For any digraph D of order n with maximum out-degree \(\Delta ^+\), it is known that \(\gamma (D)\ge \lceil n/(\Delta ^++1)\rceil \). We show that \(\gamma (D)\ge s\ell (D)\ge \lceil n/(\Delta ^++1)\rceil \) and the difference between \(s\ell (D)\) and \(\lceil n/(\Delta ^++1)\rceil \) can be arbitrarily large. In particular, for an oriented tree T of order n with \(n_0\) vertices of out-degree 0, we show that \((n-n_0+1)/2\le s\ell (T)\le \gamma (T)\le 2s\ell (T)-1\) and moreover, each value between the lower bound \(s\ell (T)\) and the upper bound \(2s\ell (T)-1\) is attainable by \(\gamma (T)\) for some oriented trees. Further, we characterize the oriented trees T for which \(s\ell (T)=(n-n_0+1)/2\) hold and show that the difference between \(s\ell (T)\) and \((n-n_0+1)/2\) can be arbitrarily large. Some other elementary properties involving the Slater number are also presented.     

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

Upper bounds on the chromatic number of triangle-free graphs with a forbidden subtree

Gyárfás conjectured that for a given forest F, there exists an integer function f(F, x) such that \(\chi (G)\le f(F,\omega (G))\) for each F-free graph G, where \(\omega (G)\) is the clique number of G. The broom B(m, n) is the tree of order \(m+n\) obtained from identifying a vertex of degree 1 of the path \(P_m\) with the center of the star \(K_{1,n}\). In this note, we prove that every connected, triangle-free and B(m, n)-free graph is \((m+n-2)\)-colorable as an extension of a result of Randerath and Schiermeyer and a result of Gyárfás, Szemeredi and Tuza. In addition, it is also shown that every connected, triangle-free, \(C_4\)-free and T-free graph is \((p-2)\)-colorable, where T is a tree of order \(p\ge 4\) and \(T\not \cong K_{1,3}\).     

Pac king spanning trees and spanning 2-connected k-edge-connected essentially $$(2 k-1)$$(2 k-1)-edge-connected subgraphs

Let \(k\ge 2, p\ge 1, q\ge 0\) be integers. We prove that every \((4kp-2p+2q)\)-connected graph contains p spanning subgraphs \(G_i\) for \(1\le i\le p\) and q spanning trees such that all \(p+q\) subgraphs are pairwise edge-disjoint and such that each \(G_i\) is k-edge-connected, essentially \((2k-1)\)-edge-connected, and \(G_i -v\) is \((k-1)\)-edge-connected for all \(v\in V(G)\). This extends the well-known result of Nash-Williams and Tutte on packing spanning trees, a theorem that every 6p-connected graph contains p pairwise edge-disjoint spanning 2-connected subgraphs, and a theorem that every \((6p+2q)\)-connected graph contains p spanning 2-connected subgraphs and q spanning trees, which are all pairwise edge-disjoint. As an application, we improve a result on k-arc-connected orientations.     

List-edge-coloring of planar graphs without 6-cycles with three chords

A graph G is edge-k-choosable if, whenever we are given a list L(e) of colors with \(|L(e)|\ge k\) for each \(e\in E(G)\), we can choose a color from L(e) for each edge e such that no two adjacent edges receive the same color. In this paper we prove that if G is a planar graph, and each 6-cycle contains at most two chords, then G is edge-k-choosable, where \(k=\max \{8,\Delta (G)+1\}\), and edge-t-choosable, where \(t=\max \{10,\Delta (G)\}\).