无$K_{1,4}$子图的图的路覆盖

For a graph G, a path cover is a set of vertex disjoint paths covering all the vertices of G, and a path cover number of G, denoted by p(G), is the minimum number of paths in a path cover among all the path covers of G. In this paper, we prove that if G is a K_(1,4)-free graph of order n and σ_(k+1)(G) ≥ n-k, then p(G) ≤ k, where σ_(k+1)(G) = min{∑v∈S d(v) : S is an independent set of G with |S| = k + 1}.     

The Criticality for Circular Coloring of Graph G_d~k

For two positive integers k and d such that k ≥ 2d, Gkd is the graph with vertex set {0,1, ...,k-1} in which ij is an edge if and only if d ≤ |i-j| ≤ k-d. Clearly, Gk1 is a complete graph of k vertices and we always assume d ≥ 2 in the following. It is easy to see (also [1]) that a graph G is (k, d)-colorable if and only if there exists a homomorphism from G to Gkd.     

Every Graph Embedded on the Surface with Euler Characteristic Number ε =-1 is Acyclically 11-choosable

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G. A graph G is acyclically k-choosable if for any list assignment L = {L(v) : v ∈ V(G)} with |L(v)| ≥ k for each vertex v ∈ V(G), there exists an acyclic proper vertex coloring φ of G such that φ(v) ∈ L(v)for each vertex v ∈ V(G). In this paper, we prove that every graph G embedded on the surface with Euler characteristic number ε =-1 is acyclically 11-choosable.     

Graphs with vertex rainbow connection number two

An edge colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. A vertex colored graph G is vertex rainbow connected if any two vertices are connected by a path whose internal vertices have distinct colors. The vertex rainbow connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G vertex rainbow connected. In 2011, Kemnitz and Schiermeyer considered graphs with rc(G) = 2.We investigate graphs with rvc(G) = 2. First, we prove that rvc(G) 2 if |E(G)|≥n-22 + 2, and the bound is sharp. Denote by s(n, 2) the minimum number such that, for each graph G of order n, we have rvc(G) 2provided |E(G)|≥s(n, 2). It is proved that s(n, 2) = n-22 + 2. Next, we characterize the vertex rainbow connection numbers of graphs G with |V(G)| = n, diam(G)≥3 and clique number ω(G) = n- s for 1≤s≤4.     

三边连通超欧拉图的一个注记

For two integers l 0 and k ≥ 0,define C(l,k) to be the family of 2-edge connected graphs such that a graph G ∈ C(l,k) if and only if for every bond S-E(G) with |S| ≤ 3,each component of G-S has order at least(|V(G)|-k)/l.In this note we prove that if a 3-edge-connected simple graph G is in C(10,3),then G is supereulerian if and only if G cannot be contracted to the Petersen graph.Our result extends an earlier result in [Supereulerian graphs and Petersen graph.JCMCC 1991,9:79-89] by Chen.     

A result on quasi k-connected graphs

Let G be a k-connected graph, and T be a subset of V(G)If G- T is not connected,then T is said to be a cut-set of GA k-cut-set T of G is a cut-set of G with |T | = kLet T be a k-cut-set of a k-connected graph GIf G- T can be partitioned into subgraphs G1 and G2 such that |G1| ≥ 2, |G2| ≥ 2, then we call T a nontrivial k-cut-set of GSuppose that G is a(k- 1)-connected graph without nontrivial(k- 1)-cut-setThen we call G a quasi k-connected graphIn this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph without K-4, then every vertex of G is incident with an edge whose contraction yields a quasi k-connected graph, and so there are at least|V(G)|2edges of G such that the contraction of every member of them results in a quasi k-connected graph.     

3,4跳图的平面性

For a graph G of size ε≥1 and its edge-induced subgraphs H1 and H2 of size r(1≤r≤ε), H1 is said to be obtained from H2 by an edge jump if there exist four distinct vertices u,v,w and x in G such that (u,v)∈E(H2), (w,x)∈ E(G)-E(H2) and H1=H2-(u,v)+(w,x). In this article, the r-jump graphs (r≥3) are discussed. A graph H is said to be an r-jump graph of G if its vertices correspond to the edge induced graph of size r in G and two vertices are adjacent if and only if one of the two corresponding subgraphs can be obtained from the other by an edge jump. For k≥2, the k-th iterated r-jump graph Jrk(G) is defined as Jr(Jrk-1(G)), where Jr1(G)=Jr(G).An infinite sequence{Gi} of graphs is planar if every graph Gi is planar. It is shown that there does not exist a graph G for which the sequence {J3k(G)} is planar, where k is any positive integer. Meanwhile,lim gen(J3k(G))=∞,where gen(G) denotes the genus of a graph G, if the sequencek→∞J3k(G) is defined for every positive integer k. As for the 4-jump gra     

Acyclic 6-choosability of Planar Graphs without 5-cycles and Adjacent 4-cycles

A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors. A graph G is acyclically k-choosable if for any list assignment L = {L(v) : v ∈ V(G)} with |L(v)| ≥ k for all v ∈ V(G), there exists a proper acyclic vertex coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V(G). In this paper, we prove that if G is a planar graph and contains no 5-cycles and no adjacent 4-cycles, then G is acyclically 6-choosable.     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

中心对称本原矩阵的k点指数集

Let G = (V, E) be a primitive digraph. The vertex exponent of G at a vertex v ∈ V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. We choose to order the vertices of G in the k-point exponent of G and is denoted by expG(k), 1 ≤ k ≤ n. We define the k-point exponent set E(n, k) := {expG(k)| G = G(A) with A ∈ CSP(n)}, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n,k) for all n, k with 1 ≤ k ≤ n except n ≡ 1(mod 2) and 1 ≤ k ≤ n - 4. We also characterize the extremal graphs when k = 1.     

外平面图的邻点可区别Ⅰ-全染色（英文）

Let G be a simple graph with no isolated edge. An Ⅰ-total coloring of a graph G is a mapping φ : V(G) ∪ E(G) → {1, 2, ···, k} such that no adjacent vertices receive the same color and no adjacent edges receive the same color. An Ⅰ-total coloring of a graph G is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, we have C_φ(u) = C_φ(v), where C_φ(u) denotes the set of colors of u and its incident edges. The minimum number of colors required for an adjacent vertex distinguishing Ⅰ-total coloring of G is called the adjacent vertex distinguishing Ⅰ-total chromatic number, denoted by χ_at~i(G).In this paper, we characterize the adjacent vertex distinguishing Ⅰ-total chromatic number of outerplanar graphs.     

环状硅酸盐图的友好指数集

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f : V(G) →Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ Z2, let vf(i) = |{v ∈ V(G) : f(v) = i}| and ef(i) = |{e ∈ E(G) : f*(e) =i}|. A labeling f of a graph G is said to be friendly if |vf(0)- vf(1)| ≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as {|ef(0)- ef(1)|: the vertex labeling f is friendly}. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS(n, m).     

关于偶正则图的R2边连通度的某些结果

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an R2-edge-cut if G-S is disconnected and contains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ″(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ″(G)=g(2k-2) for any 2k-regular connected graph G(≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.     

A Note on L（2, 1）-labelling of Trees

An L(2,1)-labelling of a graph G is a function from the vertex set V (G) to the set of all nonnegative integers such that |f(u) f(v)| ≥ 2 if d G (u,v)=1 and |f(u) f(v)| ≥ 1 if d G (u,v)=2.The L(2,1)-labelling problem is to find the smallest number,denoted by λ(G),such that there exists an L(2,1)-labelling function with no label greater than it.In this paper,we study this problem for trees.Our results improve the result of Wang [The L(2,1)-labelling of trees,Discrete Appl.Math.154 (2006) 598-603].     

INDEPENDENT-SET-DELETABLE FACTOR-CRITICAL POWER GRAPHS

It is said that a graph G is independent-set-deletable factor-critical (in short, ID-factor-critical), if, for every independent set 7 which has the same parity as |V(G)|, G-I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The k-th power of G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G3 and T(G) (the total graph of G) are ID-factor-critical, and G4 (when |V(G)| is even) is strongly IM-extendable; if G is 2-connected, then D2 is ID-factor-critical.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

A conjecture on k-factor-critical and 3-γ-critical graphs

For a graph G =(V,E),a subset VS is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S.The domination number γ(G) of G is the minimum order of a dominating set in G.A graph G is said to be domination vertex critical,if γ(G-v) γ(G) for any vertex v in G.A graph G is domination edge critical,if γ(G ∪ e) γ(G) for any edge e ∈/E(G).We call a graph G k-γ-vertex-critical(resp.k-γ-edge-critical) if it is domination vertex critical(resp.domination edge critical) and γ(G) = k.Ananchuen and Plummer posed the conjecture:Let G be a k-connected graph with the minimum degree at least k+1,where k 2 and k≡|V|(mod 2).If G is 3-γ-edge-critical and claw-free,then G is k-factor-critical.In this paper we present a proof to this conjecture,and we also discuss the properties such as connectivity and bicriticality in 3-γ-vertex-critical claw-free graph.     

Heavy cycles in 2-connected triangle-free weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V(G), then G contains a cycle of weight at least 2d.     

1-planar Graphs without 4-cycles or 5-cycles are 5-colorable

A graph is 1-planar if it can be drawn on the Euclidean plane so that each edge is crossed by at most one other edge. A proper vertex k-coloring of a graph G is defined as a vertex coloring from a set of k colors such that no two adjacent vertices have the same color. A graph that can be assigned a proper k-coloring is k-colorable. A cycle is a path of edges and vertices wherein a vertex is reachable from itself. A cycle contains k vertices and k edges is a k-cycle. In this paper, it is proved t...     

The 2-pebbling property for dense graphs

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f(G) is the smallest number m such that for every distribution of m pebbles and every vertex v,a pebble can be moved to v. A graph G is said to have the 2-pebbling property if for any distribution with more than 2f(G) q pebbles, where q is the number of vertices with at least one pebble, it is possible,using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G(s,t) has the 2-pebbling property, where G(s, t) is a bipartite graph with partite sets of size s and t (s ≥ t). Similarly, the-pebbling number f (G) is the smallest number m such that for every distribution of m pebbles and every vertex v, pebbles can be moved to v. Herscovici et al. conjectured that f(G) ≤ 1.5n + 8-6 for the graph G with diameter 3, where n = |V (G)|. In this paper, we prove that if s ≥ 15 and G(s, t) has minimum degree at least (s+1)/ 2 , then f (G(s, t)) = s + t, G(s, t) has the 2-pebbling property and f (G(s, t)) ≤ s + t + 8(-1). In other words, we extend a result due to Czygrinow and Hurlbert, and show that the above Snevily conjecture and Herscovici et al. conjecture are true for G(s, t) with s ≥ 15 and minimum degree at least (s+1)/ 2 .