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.     

A special k-coloring for a connected k-chromatic graph

For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number χ(G) = k can be properly vertex colored by k colors so that for each pair of vertices x_o and x_p in any color class there exist vertices x₁, x₂, …, x_p-1 of the same class with dist(x_i, x_i+1) f(k) for each i, 0 i p − 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.

We prove that f(k) < 12k when the order of the graph is k(k − 2) + 1.     

Multicolor routing in the undirected hypercube

An undirected routing problem is a pair (G,R) where G is an undirected graph and R is an undirected multigraph such that V(G)=V(R). A solution to an undirected routing problem (G,R) is a collection P of undirected paths of G (possibly containing multiple occurrences of the same path) such that edges of R are in one-to-one correspondence with the paths of P, with the path corresponding to edge {u,v} connecting u and v. We say that a collection of paths P is k-colorable if each path of P can be colored by one of the k colors so that the paths of the same color are edge-disjoint (each edge of G appears at most once in the paths of each single color). In the circuit-switched routing context, and in optical network applications in particular, it is desirable to find a solution to a routing problem that is colorable with as few colors as possible. Let Q_n denote the n-dimensional hypercube, for arbitrary n1. We show that a routing problem (Q_n,R) always admits a 4d-colorable solution where d is the maximum vertex degree of R. This improves over the 16d/2-color result which is implicit in the previous work of Aumann and Rabani [SODA95, pp. 567–576]. Since, for any positive d, there is a multigraph R of degree d such that any solution to (Q_n,R) requires at least d colors, our result is tight up to a factor of four. In fact, when d=1, it is tight up to a factor of two, since there is a graph of degree one (the antipodal matching) that requires two colors.     

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.     

Multicolored forests in complete bipartite graphs

If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=(a₁,a₂,…,a_k), in which a_i denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph K_n,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of K_n,n into multicolored trees.     

Cyclic coloring of plane graphs

Let G be a plane graph, and let χ_k(G) be the minimum number of colors to color the vertices of G so that every two of them which lie in the boundary of the same face of the size at most k, receive different colors. In 1966, Ore and Plummer proved that χ_k(G)2k for any k3. It is also known that χ₃(G)4 (Appel and Haken, 1976) and χ₄(G)6 (Borodin, 1984). The result in the present paper is: χ₅(G)9, χ₆(G)11, χ₇(G)12, and χ_k(G)2k − 3 if k8.     

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.     

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.     

The connectivity of a graph on uniform points on [0,1]

A random graph G_n(x) is constructed on independent random points U₁,…,U_n distributed uniformly on [0,1]^d, d1, in which two distinct such points are joined by an edge if the l_∞-distance between them is at most some prescribed value 0<x<1. The connectivity distance c_n, the smallest x for which G_n(x) is connected, is shown to satisfy

(1)

For d2, the random graph G_n(x) behaves like a d-dimensional version of the random graphs of Erdös and Rényi, despite the fact that its edges are not independent: c_n/d_n→1, a.s., as n→∞, where d_n is the largest nearest-neighbor link, the smallest x for which G_n(x) has no isolated vertices.     

Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V (G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …,|E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to (1+|E(G)|)/2·d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V (G),|{e:e ∈ E_v, f(e) ≤ Δ/2}|=|{e:e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.     

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 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.     

A note on G-intersecting families

Consider a graph G and a k-uniform hypergraph on common vertex set [n]. We say that is G-intersecting if for every pair of edges in there are vertices xX and yY such that x=y or x and y are joined by an edge in G. This notion was introduced by Bohman, Frieze, Ruszinkó and Thoma who proved a natural generalization of the Erd s–Ko–Rado Theorem for G-intersecting k-uniform hypergraphs for G sparse and k=O(n^1/4). In this note, we extend this result to .     

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).     

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.     

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).     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

Precoloring extension. I. Interval graphs

This paper is the first article in a series devoted to the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this ‘precoloring’ be extended to a proper coloring of G with at most k colors (for some given k)? This question was motivated by practical problems in scheduling and VLSI theory. Here we investigate its complexity status for interval graphs and for graphs with a bounded treewidth.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.