Factors and Connected Induced Subgraphs

 Let G be a connected graph without loops and without multiple edges, and let p be an integer such that 0 < p<|V(G)|. Let f be an integer-valued function on V(G) such that 2≤f(x)≤ deg _G (x) for all x∈V(G). We show that if every connected induced subgraph of order p of G has an f-factor, then G has an f-factor, unless ∑ _{x ∈ V ( G )} f(x) is odd. Received: June 29, 1998?Final version received: July 30, 1999     

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.     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

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     

Average distance,minimum degree,and spanning trees

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ()⁻¹ Σ_{x,y}⊂V(G) d_G(x, y), where V(G) denotes the vertex set of G and d_G(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most . We give improved bounds for K₃‐free graphs, C₄‐free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 1–13, 2000     

Degree conditions and degree bounded trees

We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σ_k(A) the minimum value of the degree sum in G of any k independent vertices in A and by w(G−A) the number of components in the induced subgraph G−A. Our main results are the following: (i) If σ_k(A)≥|V(G)|−1, then G contains a tree T with maximum degree at most k and A⊆V(T). (ii) If σ_k−w(G−A)(A)≥|A|−1, then G contains a spanning tree T such that d_T(x)≤k for every x∈A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.     

<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.     

Combinatorial interpretations for TG(1, −1)

Let G be a connected and simple graph with vertex set {1, 2, …, n + 1} and T_G(x, y) the Tutte polynomial of G. In this paper, we give combinatorial interpretations for T_G(1, ?1). In particular, we give the definitions of even spanning tree and left spanning tree. We prove T_G(1, ?1) is the number of even‐left spanning trees of G. We associate a permutation with a spanning forest of G and give the definition of odd G‐permutations. We show T_G(1, ?1) is the number of odd G‐permutations. We give a bijection from the set of odd K_{n + 1}‐permutations to the set of alternating permutations on the set {1, 2, …, n}. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 341–348, 2012     

A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V(G)and an edge set E(G). A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1}.The weight of f is W(f)=∑_(x∈V)(G)∪E(G))f(X).For an element x∈V(G)∪E(G),we define f[x]=∑_(y∈NT[x])f(y).A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1} such that f[x]≥1 for all x∈V(G)∪E(G).The total signed domination numberγ_s~*(G)of G is the minimum weight of a total signed domination function on G. In this paper,we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values ofγ_s~*(G)when G is C_n and P_n.     

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x ₁, x ₂, . . . , x _n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x ₁, x ₂, . . . , x _n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.     

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     

Signed 2-independence in graphs

A function f : V→{−1,1} defined on the vertices of a graph G=(V,E) is a signed 2-independence function if the sum of its function values over any closed neighbourhood is at most one. That is, for every vV, f(N[v])1, where N[v] consists of v and every vertex adjacent to v. The weight of a signed 2-independence function is f(V)=∑f(v), over all vertices vV. The signed 2-independence number of a graph G, denoted α_s²(G), equals the maximum weight of a signed 2-independence function of G. In this paper, we establish upper bounds for α_s²(G) in terms of the order and size of the graph, and we characterize the graphs attaining these bounds. For a tree T, upper and lower bounds for α_s²(T) are established and the extremal graphs characterized. It is shown that α_s²(G) can be arbitrarily large negative even for a cubic graph G.     

A new domination conception

Let f be an integer valued function defined on the vertex set V(G) of a simple graph G. We call a subset D_f of V(G) a f-dominating set of G if |N(x, G) ∩ D_f| ≥ f(x) for all x ∈ V(G) — D_f, where N(x, G) is the set of neighbors of x. D_f is a minimum f-dominating set if G has no f-dominating set D′_f with |D_f| < |D_f|. If j, k ∈ N₀ = {0,1,2,…} with j ≤ k, then we define the integer valued function f_j,k on V(G) by . By μ_j,k(G) we denote the cardinality of a minimum f_j,k-dominating set of G. A set D ? V(G) is j-dominating if every vertex, which is not in D, is adjacent to at least j vertices of D. The j-domination number γ_j(G) is the minimum order of a j-dominating set in G. In this paper we shall give estimations of the new domination number μ_j,k(G), and with the help of these estimations we prove some new and some known upper bounds for the j-domination number. © 1993 John Wiley & Sons, Inc.     

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     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

T-Colorings and T-Edge Spans of Graphs

 Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C ^d _n of the n-cycle C _n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C _n ^d for n≡0 or (mod d+1), and lower and upper bounds for other cases. Received: May 13, 1996 Revised: December 8, 1997     

Rooted induced trees in triangle‐free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceilFor a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceil$, improving a recent result by Fox, Loh and Sudakov. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206–209, 2010     

Further results on the lower bounds of mean color numbers

Let G be a graph with n vertices. The mean color number of G, denoted by μ(G), is the average number of colors used in all n‐colorings of G. This paper proves that μ(G) ≥ μ(Q), where Q is any 2‐tree with n vertices and G is any graph whose vertex set has an ordering x₁,x₂,…,x_n such that x_i is contained in a K₃ of G[V_i] for i = 3,4,…,n, where V_i = {x₁,x₂,…,x_i}. This result improves two known results that μ(G) ≥ μ(O_n) where O_n is the empty graph with n vertices, and μ(G) ≥ μ(T) where T is a spanning tree of G. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 51–73, 2005     

Inequalities involving independence domination, f-domination, connected and total f-domination numbers

Let f be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an f-dominating set if each vertex x outside D is adjacent to at least f(x) vertices in D. The minimum number of vertices in an f-dominating set is defined to be the f-domination number, denoted by _f (G). In a similar way one can define the connected and total f-domination numbers _c,f (G) and _t,f (G). If f(x) = 1 for all vertices x, then these are the ordinary domination number, connected domination number and total domination number of G, respectively. In this paper we prove some inequalities involving _f (G), _c,f (G), _t,f (G) and the independence domination number i(G). In particular, several known results are generalized.