Minimum Cost Homomorphism Dichotomy for Oriented Cycles

For digraphs D and H, a mapping f : V(D) → V(H) is a homomorphism of D to H if uv ∈ A(D) implies f(u) f(v) ∈ A(H). If, moreover, each vertex u ∈ V(D) is associated with costs c _i (u), i ∈ V(H), then the cost of the homomorphism f is ∑ _{u ∈V(D)} c _f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H (abbreviated MinHOM(H)). The problem is to decide, for an input graph D with costs c _i (u), u ∈ V(D), i ∈ V(H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. We obtain a dichotomy classification for the time complexity of MinHOM(H) when H is an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops.     

Group Chromatic Number of Graphs without K5-Minors

 Let G be a graph with a fixed orientation and let A be a group. Let F(G,A) denote the set of all functions f: E(G) ↦A. The graph G is A -colorable if for any function f∈F(G,A), there is a function c: V(G) ↦A such that for every directed e=u v∈E(G), c(u)−c(v)≠f(e). The group chromatic numberχ₁(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m under a given orientation D. In [J. Combin. Theory Ser. B, 56 (1992), 165–182], Jaeger et al. proved that if G is a simple planar graph, then χ₁(G)≤6. We prove in this paper that if G is a simple graph without a K ₅-minor, then χ₁(G)≤5. Received: August 18, 1999 Final version received: December 12, 2000     

Bounds on the Distance Two-Domination Number of a Graph

 For a graph G = (V, E), a subset D⊆V(G) is said to be distance two-dominating set in G if for each vertex u∈V−D, there exists a vertex v∈D such that d(u,v)≤2. The minimum cardinality of a distance two-dominating set in G is called a distance two-domination number and is denoted by γ₂(G). In this note we obtain various upper bounds for γ₂(G) and characterize the classes of graphs attaining these bounds. Received: May 31, 1999 Final version received: July 13, 2000     

Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑_{x,y}⊆V d(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑ _e=uv∈E n _u (e|G)n _v (e|G), where n _u (e|G) (resp. n _v (e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑ _e=uv∈E [n _eu (e|G)+n _ev (e|G)], where n _eu (e|G) (resp. n _ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.     

On Connected [g,f +1]-Factors in Graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z⁺-{1} such that g(v) ≤ f(v) ≤ d_G(v) for all v ∈V (G), where d_G(v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs. * The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025. † The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).     

<Emphasis Type="Italic">f</Emphasis>-Colorings of some graphs of <Emphasis Type="Italic">f</Emphasis>-class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v V（G） at most f（v） times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by X′f（G）. Any simple graph G has the f-chromatic index equal to △f（G） or △f（G） ＋ 1, where △f（G） =max v V（G）{[d（v）/f（v）]}. If X′f（G） = △f（G）, then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with △f（G） colors are given.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

A Generalization of the Gallai–Roy Theorem

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex v∈V(G), there is a path P starting at v which represents all χ(G) colors. Received: May 20, 1999 Final version received: December 24, 1999     

On vertex-coloring edge-weighting of graphs

A k-edge-weighting w of a graph G is an assignment of an integer weight, w(e) ∈ {1,…,k}, to each edge e. An edge-weighting naturally induces a vertex coloring c by defining c(u) = Σ _e∋u w(e) for every u ∈ V (G). A k-edge-weighting of a graph G is vertex-coloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). When k ≡ 2 (mod 4) and k ⩾ 6, we prove that if G is k-colorable and 2-connected, δ(G) ⩾ k − 1, then G admits a vertex-coloring k-edge-weighting. We also obtain several sufficient conditions for graphs to be vertex-coloring k-edge-weighting.      

Group connectivity and group colorings of graphs — A survey

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A − {0}. The graph G is A-connected if G has an orientation D(G) such that for every map b: V (G) ↦ A satisfying Σ _v∈V(G) b(v) = 0, there is a function f: E(G) ↦ A* such that for each vertex v ∈ V (G), the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b(v). The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections.

The geodetic numbers of graphs and digraphs

For every two vertices u and v in a graph G,a u-v geodesic is a shortest path between u and v.Let I(u,v)denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I(S) denote the union of all I(u,v)for u,v∈S.The geodetic number g(G)of a graph G is the minimum cardinality of a set S with I(S)=V(G).For a digraph D,there is analogous terminology for the geodetic number g(D).The geodetic spectrum of a graph G,denoted by S(G),is the set of geodetic numbers of all orientations of graph G.The lower geodetic number is g~-(G)=minS(G)and the upper geodetic number is g~ (G)=maxS(G).The main purpose of this paper is to study the relations among g(G),g~-(G)and g~ (G)for connected graphs G.In addition,a sufficient and necessary condition for the equality of g(G)and g(G×K_2)is presented,which improves a result of Chartrand,Harary and Zhang.     

Orthogonal factorizations of digraphs

Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g ⁻, g ⁺) and ƒ = (ƒ ⁻, ƒ ⁺) be pairs of positive integer-valued functions defined on V(G) such that g ⁻(x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ ƒ ⁺(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g ⁻(x) ⩽ id _H (x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ od _H (x) ⩽ ƒ ⁺(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let = {F ₁, F ₂,…, F _m} and H be a factorization and a subdigraph of G, respectively. is called k-orthogonal to H if each F _i , 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g ⁻(x), g ⁺(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.      

A New Result on Alspach's Problem

Let G be a simple graph. Let g(x) and f(x) be integer-valued functions defined on V(G) with g(x)≥2 and f(x)≥5 for all x∈V(G). It is proved that if G is an (mg+m−1, mf−m+1)-graph and H is a subgraph of G with m edges, then there exists a (g,f)-factorization of G orthogonal to H. Received: January 19, 1996 Revised: November 11, 1996     

An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

Decomposition of Graphs into (g, f)-Factors

Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg _G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results. Revised: June 11, 1998     

A Generalization of Orthogonal Factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d _H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let = {F ₁, F ₂, ..., F _m } be a factorization of G and H be a subgraph of G with mr edges. If F _i , 1 ≤i≤m, has exactly r edges in common with H, then is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges. This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China     

Counting characters in linear group actions

Let G be a finite group and V be a finite G-module. We present upper bounds for the cardinalities of certain subsets of Irr(GV), such as the set of those χ ∈ Irr(GV) such that, for a fixed v ∈ V, the restriction of χ to 〈v〉 is not a multiple of the regular character of 〈v〉. These results might be useful in attacking the noncoprime k(GV)-problem.     

Implicit Euler scheme for an abstract evolution inequality

For a triple {V, H, V*} of Hilbert spaces, we consider an evolution inclusion of the form u′(t)+A(t)u(t)+δϕ(t, u(t)) ∋ f(t), u(0) = u0, t ∈ (0, T ], where A(t) and ϕ(t, ·), t ∈ [0, T], are a family of nonlinear operators from V to V * and a family of convex lower semicontinuous functionals with common effective domain D(ϕ) ⊂ V. We indicate conditions on the data under which there exists a unique solution of the problem in the space H ¹(0, T; V)∩W _∞¹ (0, T;H) and the implicit Euler method has first-order accuracy in the energy norm.     

On super edge-antimagic total labelings of

A graph G of order p and size q is called (a,d)-edge-antimagic total if there exists a bijective function f:V(G)E(G)→{1,2,…,p+q} such that the edge-weights w(uv)=f(u)+f(v)+f(uv), uvE(G), form an arithmetic sequence with first term a and common difference d. The graph G is said to be super (a,d)-edge-antimagic total if the vertex labels are 1,2,…,p. In this paper we study super (a,d)-edge-antimagic properties of mK_n, that is, of the graph formed by the disjoint union of m copies of K_n.