Eulerian subgraphs containing given edges

For an integer l0, define to be the family of graphs such that if and only if for any edge subset XE(G) with |X|l, G has a spanning eulerian subgraph H with XE(H). The graphs in are known as supereulerian graphs. Let f(l) be the minimum value of k such that every k-edge-connected graph is in . Jaeger and Catlin independently proved f(0)=4. We shall determine f(l) for all values of l0. Another problem concerning the existence of eulerian subgraphs containing given edges is also discussed, and former results in [J. Graph Theory 1 (1977) 79–84] and [J. Graph Theory 3 (1979) 91–93] are extended.     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).     

Vertex pancyclic graphs

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3kn, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3kn. In this paper, we shall present different sufficient conditions for graphs to be vertex pancyclic.     

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.     

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.     

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.     

On approximation intractability of the path–distance–width problem

Path–distance–width of a graph G=(V,E), denoted by pdw(G), is the minimum integer k satisfying that there is a nonempty subset of SV such that the number of the nodes with distance i from S is at most k for any nonnegative integer i. It is known that given a positive integer k and a graph G, the decision problem pdw(G)k is NP-complete even if G is a tree (Yamazaki et al. Lecture Notes in Computer Science, vol. 1203, Springer, Berlin, 1997, pp. 276–287). In this paper, we show that it is NP-hard to approximate the path–distance–width of a graph within a ratio for any >0, even for trees.     

Acyclic coloring of graphs without bichromatic long path

Let G be a graph of maximum degree Δ. A proper vertex coloring of G is acyclic if there is no bichromatic cycle. It was proved by Alon et al. [Acyclic coloring of graphs. Random Structures Algorithms, 1991, 2(3): 277−288] that G admits an acyclic coloring with O(Δ^4/3) colors and a proper coloring with O(Δ^{(k−1)/(k−2)}) colors such that no path with k vertices is bichromatic for a fixed integer k≥5. In this paper, we combine above two colorings and show that if k≥5 and G does not contain cycles of length 4, then G admits an acyclic coloring with O(Δ^{(k−1)/(k−2)}) colors such that no path with k vertices is bichromatic.     

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.     

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.     

Choosability of planar graphs

A graph G = G(V, E) with lists L(v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L(v). We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with L(v) = k v V(G).

Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.     

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.     

Networks with small stretch number

In a previous work, the authors introduced the class of graphs with bounded induced distance of order k (BID(k) for short), to model non-reliable interconnection networks. A network modeled as a graph in BID(k) can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is at most k times the distance in the non-faulty graph. The smallest k such that GBID(k) is called stretch number of G. We show an odd characteristic of the stretch numbers: every rational number greater or equal 2 is a stretch number, but only discrete values are admissible for smaller stretch numbers. Moreover, we give a new characterization of classes BID(2−1/i), i1, based on forbidden induced subgraphs. By using this characterization, we provide a polynomial time recognition algorithm for graphs belonging to these classes, while the general recognition problem is Co-NP-complete.     

k-Subdomination in graphs

For a positive integer k, a k-subdominating function of a graph G=(V,E) is a function f : V→{−1,1} such that ∑_uNG[v]f(u)1 for at least k vertices v of G. The k-subdomination number of G, denoted by γ_ks(G), is the minimum of ∑_vVf(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We also give a lower bound for γ_ks(G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number γ_ks(G), the signed domination number γ_s(G) and the majority domination number γ_maj(G).     

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.     

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.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

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.     

Stable 2-pairs and (X,Y)-intersection graphs

Given two fixed graphs X and Y, the (X,Y)-intersection graph of a graph G is a graph where

1. each vertex corresponds to a distinct induced subgraph in G isomorphic to Y, and

2. two vertices are adjacent iff the intersection of their corresponding subgraphs contains an induced subgraph isomorphic to X.

This notion generalizes the classical concept of line graphs since the (K₁,K₂)-intersection graph of a graph G is precisely the line graph of G.

Let ( , respectively) denote the family of line graphs of bipartite graphs (bipartite multigraphs, respectively), and refer to a pair (X,Y) as a 2-pair if Y contains exactly two induced subgraphs isomorphic to X. Then and , respectively, are the smallest families amongst the families of (X,Y)-intersection graphs defined by so called hereditary 2-pairs and hereditary non-compact 2-pairs. Furthermore, they can be characterized through forbidden induced subgraphs. With this motivation, we investigate the properties of a 2-pair (X,Y) for which the family of (X,Y)-intersection graphs coincides with (or ). For this purpose, we introduce a notion of stability of a 2-pair and obtain the desired characterization for such stable 2-pairs. An interesting aspect of the characterization is that it is based on a graph determined by the structure of (X,Y).     

Optimally super-edge-connected transitive graphs

Let X=(V,E) be a connected regular graph. X is said to be super-edge-connected if every minimum edge cut of X is a set of edges incident with some vertex. The restricted edge connectivity λ′(X) of X is the minimum number of edges whose removal disconnects X into non-trivial components. A super-edge-connected k-regular graph is said to be optimally super-edge-connected if its restricted edge connectivity attains the maximum 2k−2. In this paper, we define the λ′-atoms of graphs with respect to restricted edge connectivity and show that if X is a k-regular k-edge-connected graph whose λ′-atoms have size at least 3, then any two distinct λ′-atoms are disjoint. Using this property, we characterize the super-edge-connected or optimally super-edge-connected transitive graphs and Cayley graphs. In particular, we classify the optimally super-edge-connected quasiminimal Cayley graphs and Cayley graphs of diameter 2. As a consequence, we show that almost all Cayley graphs are optimally super-edge-connected.