Cycle-saturated graphs of minimum size

A graph G is called C_k-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in C_k-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + c₁n/k and n + c₂n/k for some positive constants c₁ and C₂. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.     

On the bounded domination number of tournaments

In a simple digraph, a star of degree t is a union of t edges with a common tail. The k-domination number γ_k(G) of digraph G is the minimum number of stars of degree at most k needed to cover the vertex set. We prove that γ_k(T)=n/(k+1) when T is a tournament with n14k lg k vertices. This improves a result of Chen, Lu and West. We also give a short direct proof of the result of E. Szekeres and G. Szekeres that every n-vertex tournament is dominated by at most lg n−lglg n+2 vertices.     

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.     

Holes in random graphs

It is shown that for every >0 with the probability tending to 1 as n→∞ a random graph G(n,p) contains induced cycles of all lengths k, 3 ≤ k ≤ (1 − )n log c/c, provided c(n) = (n − 1)p(n)→∞.     

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.     

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.     

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.     

Nowhere-zero 3-flows of highly connected graphs

Let G be a k-edge-connected graph of order n. If k4log₂ n then G has a nowhere-zero 3-flow.     

Number of minimum vertex cuts in transitive graphs

Let G be a k-regular vertex transitive graph with connectivity κ(G)=k and let m_k(G) be the number of vertex cuts with k vertices. Define m(n,k)=min{m_k(G): GT_n,k}, where T_n,k denotes the set of all k-regular vertex transitive graphs on n vertices with κ(G)=k. In this paper, we determine the exact values of m(n,k).     

The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f₁,f₂,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size f_i. It is proved that there exist constants c=c(n) and k₀=k₀(n) such that f_k+1=f_k+f_k−c for all k>k₀, and lim_n→∞ c(n)/(nlogn)=1.     

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.     

带松弛条件的图的强边着色

给定两个非负整数s和t,图G的(s,t)-松弛强k边着色可表示为映射c:E(G)→[k],这个映射满足对G中的任意一条边e,颜色c(e)在e的1-邻域中最多出现s次并且在e的2-邻域中最多出现t次.图G的(s,t)-松弛强边着色指数,记作x'(s,t)(G),表示使得图G有(s,t)-松弛强k边着色的最小k值.在图G中...     

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.     

No-hole L(2,1)-colorings

An L(2,1)-coloring of a graph G is a coloring of G's vertices with integers in {0,1,…,k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…,k}.

Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.     

Measurements of edge-uncolorability

Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k=2,3, let c_k be the maximum size of a k-colorable subgraph of a cubic graph G=(V,E). We consider r₃=|E|−c₃ and . We show that on one side r₃ and r₂ bound each other, but on the other side that the difference between them can be arbitrarily large. We also compare them to the oddness ω of G, the smallest possible number of odd circuits in a 2-factor of G. We construct cyclically 5-edge connected cubic graphs where r₃ and ω are arbitrarily far apart, and show that for each 1c<2 there is a cubic graph such that ωcr₃. For k=2,3, let ζ_k denote the largest fraction of edges that can be k-colored. We give best possible bounds for these parameters, and relate them to each other.     

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.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

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.     

Matching extension and minimum degree

Let G be a simple connected graph on 2n vertices with a perfect matching. For a given positive integer k, 1 k n − 1, G is k-extendable if for every matching M of size k in G, there exists a perfect matching in G containing all the edges of M. The problem that arises is that of characterizing k-extendable graphs. In this paper, we establish a necessary condition, in terms of minimum degree, for k-extendable graphs. Further, we determine the set of realizable values for minimum degree of k-extendable graphs. In addition, we establish some results on bipartite graphs including a sufficient condition for a bipartite graph to be k-extendable.     

Pancyclicity mod k of claw-free graphs and K1,4-free graphs

For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K_1,4-free graph G with minimum degree δ(G)k+3 is pancyclic mod k and every claw-free graph G with δ(G)k+1 is pancyclic mod k, which confirms Thomassen's conjecture (J. Graph Theory 7 (1983) 261–271) for claw-free graphs.