Covering vertices of a graph by k disjoint cycles

Let d, k and n be three integers with k3, d4k−1 and n3k. We show that if d(x)+d(y)d for each pair of nonadjacent vertices x and y of a graph G of order n, then G contains k vertex-disjoint cycles converting at least min{d,n} vertices of G.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Reconstruction of infinite locally finite connected graphs

Let G be an infinite locally finite connected graph. We study the reconstructibility of G in relation to the structure of its end set . We prove that an infinite locally finite connected graph G is reconstructible if there exists a finite family (Ω_i)_0i (n2) of pairwise finitely separable subsets of such that, for all x,y,x′,y′V(G) and every isomorphism f of G−{x,y} onto G−{x′,y′} there is a permutation π of {0,…,n−1} such that for 0i<n. From this theorem we deduce, as particular consequences, that G is reconstructible if it satisfies one of the following properties: (i) G contains no end-respecting subdivision of the dyadic tree and has at least two ends of maximal order; (ii) the set of thick ends or the one of thin ends of G is finite and of cardinality greater than one. We also prove that if almost all vertices of G are cutvertices, then G is reconstructible if it contains a free end or if it has at least a vertex which is not a cutvertex.     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

Explicit formulas for the Dumont-Foata polynomial

Dumont and Foata have defined a polynomial F_n(x, y, z) recursively. They proved that F_n(x, y, z) is symmetric in x, y, z and that F_n(1, 1, 1) = G_2n+2 the Genocchi number. Moreover, they gave an elegant combinatorial interpretation for the coefficients of F_n(x, y, z). In the present paper explicit formulas and generating functions for F_n(x, y, z) are obtained.     

Maximum number of edges in a critically k-connected graph

A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any vV(G). We show that if n,k are integers with k4 and nk+2, and G is a critically k-connected graph of order n, then |E(G)|n(n−1)/2−p(n−k)+p²/2, where p=(n/k)+1 if n/k is an odd integer and p=n/k otherwise. We also characterize extremal graphs.     

Some Ore-type Results for Matching and Perfect Matching in <Emphasis Type="Italic">k</Emphasis>-uniform Hypergraphs

Let S₁ and S₂ be two (k-1)-subsets in a k-uniform hypergraph H. We call S₁ and S₂ strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S₁ ∩ e ≠∅ and S₂ ∩ e ≠∅ or e ⊆ S₁ ∪ S₂ or e ⊇ S₁ ∪ S₂, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k²-k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k-1)-sets equal to 2n-4(k-1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k-2)k. If the degree sum of any two middle independent (k-1)-subsets is larger than 2(d-1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k-1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.     

The k-ball l-path branch weight centroid

If x is a vertex of a tree T of radius r, if k and l are integers, if 0 k r, 0 l r, and if P is an l-path with one end at x, then define β(x; k, P) to be the number of vertices of T that are reachable from x via the l-path P and that are outside of the k-ball about x. That is, β(x;k,P) = {yεV(T):y is reachable from x via P,d(x,y) > k}. Define the k-ball l-path branch weight of x, denoted β(x;k,l), to be max {β(x;k,P):P an l-path with one end at x}, and define the k-balll-path branch weight centroid of T, denoted B(T;k,l), to be the set xεV(T): β(x;k,l) β(y;k,l), yεV(T). This two-parameter family of central sets in T includes the one-parameter family of central sets called the k-nuclei introduced by Slater (1981) which has been shown to be the one parameter family of central sets called the k-branch weight centroids by Zaw Win (1993). It also includes the one-parameter family of central sets called the k-ball branch weight centroid introduced by Reid (1991). In particular, this new family contains the classical central sets, the center and the median (which Zelinka (1968) showed is the ordinary branch weight centroid). The sets obtained for particular values of k and l are examined, and it is shown that for many values they consist of one vertex or two adjacent vertices.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

Inverse geophysical problems for some noncompactly supported inhomogeneities

Let[²+k²n(x₁,x₃)]u(x₁,x₂,x₃)=−δ(x₁,y₁δ(x₂,y₂)δ(x₃,y₃) in R³₊. Assume that u(x₁,x₂,x₃=0,y₁,y₂=0,y₃=0,k) is measured at the plane P {x:x₃=0} for all positions of the source on the line y = (y₁,y₂ = 0,y₃ = 0), -∞ < y₁ < ∞, and receiver on the plane(x₁,x₂,x₃ − <x₁,x₂ < ∞, and for low-frequencies 0 < k <k₀, k₀ > 0 is an arbitrary small wave number. Assume thatn(x₁,x₃) is an arbitrary bounded piecewise-continuous function. The basic result is: the above low-frequency surface data determinen(x₁,x₃)uniquely.     

Hamiltonian cycles in n-factor-critical graphs

A graph G is said to be n-factor-critical if G−S has a 1-factor for any SV(G) with |S|=n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with , then G is hamiltonian with some exceptions. To extend this theorem, we define a (k,n)-factor-critical graph to be a graph G such that G−S has a k-factor for any SV(G) with |S|=n. We conjecture that if G is a 2-connected (k,n)-factor-critical graph of order p with , then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k2.     

Independent sets and repeated degrees

We answer a question of Erdös, Faudree, Reid, Schelp and Staton by showing that for every integer k 2 there is a triangle-free graph G of order n such that no degree in G is repeated more than k times and ind(G) = (1 + o(1))n/k.     

路的字典积的邻和可区别边染色

图G的正常[k]-边染色σ是指颜色集合为[k]={1,2,…,k}的G的一个正常边染色.用w_σ（x）表示顶点x关联边的颜色之和,即w_σ（x）=∑_e∋x σ（e）,并称w_σ（x）关于σ的权.图G的k-邻和可区别边染色是指相邻顶点具有不同权的正常[k]-边染色,最小的k值称为G的邻和可区别边色数,记为χ'_∑（G）.现得到了路P_n与简单连通图H的字典积P_n[H]的邻和可区别边色数的精确值,其中H分别为正则第一类图、路、完全图的补图.     

Primitive graphs with given exponents and minimum number of edges

A graph G = (V, E) on n vertices is primitive if there is a positive integer k such that for each pair of vertices u, v of G, there is a walk of length k from u to v. The minimum value of such an integer, k, is the exponent, exp(G), of G. In this paper, we find the minimum number, h(n, k), of edges of a simple graph G on n vertices with exponent k, and we characterize all graphs which have h(n, k) edges when k is 3 or even.     

Maximal IM-unextendable graphs

A graph G is maximal IM-unextendable if G is not induced matching extendable and, for every two nonadjacent vertices x and y, G+xy is induced matching extendable. We show in this paper that a graph G is maximal IM-unextendable if and only if G is isomorphic to M_r(K_s(K_n1K_n2K_nt)), where M_r is an induced matching of size r, r1, t=s+2, and each n_i is odd.     

A Note on Strong Edge Coloring of Sparse Graphs

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ'_s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2 k. Yang and Zhu [J. Graph Theory, 83, 334–339(2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4 kΔ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤ 4 kΔ-2 k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most(4 k-1)Δ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤(4 k-1)Δ-2 k + 1.     

A note on minimum degree conditions for supereulerian graphs

A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut SE(G) with |S|3 satisfies the property that each component of G−S has order at least (n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4: If G is a 2-edge-connected graph of order n with δ(G)4 such that for every edge xyE(G) , we have max{d(x),d(y)}(n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ(G)4 cannot be relaxed.     

Graphs with prescribed local connectivities

A graph G is locally n-connected (locally n-edge connected) if the neighborhood of each vertex of G is n-connected (n-edge connected). The local connectivity (local edge-connectivity) of G is the maximum n for which G is locally n-connected (locally n-edge connected). It is shown that if k and m are integers with O k < m, then a graph exists which has connectivity m and local connectivity k. Furthermore, such a graph with smallest order is determined. Corresponding results are obtained involving the local connectivity and the local edge-conectivity.