A Havel-Hakimi type procedure and a sufficient condition for a sequence to be potentially S r,s -graphic

The split graph K _r + $\overline {{K_s}} $ on r+s vertices is denoted by S _r,s . A non-increasing sequence π = (d ₁, d ₂, …, d _n ) of nonnegative integers is said to be potentially S _r,s -graphic if there exists a realization of π containing S _r,s as a subgraph. In this paper, we obtain a Havel-Hakimi type procedure and a simple sufficient condition for π to be potentially S _r,s -graphic. They are extensions of two theorems due to A.R.Rao (The clique number of a graph with given degree sequence, Graph Theory, Proc. Symp., Calcutta 1976, ISI Lect. Notes Series 4 (1979), 251–267 and An Erd?s-Gallai type result on the clique number of a realization of a degree sequence, unpublished).     

Degree sequences of graphs containing a cycle with prescribed length

Let r ≥ 3, n ≥ r and π = (d ₁, d ₂, ..., d _n ) be a non-increasing sequence of nonnegative integers. If π has a realization G with vertex set V (G) = {v ₁, v ₂, ..., v _n } such that d _G (v _i ) = d _i for i = 1, 2, ..., n and v ₁ v ₂ ... v _r v ₁ is a cycle of length r in G, then π is said to be potentially C _r ″-graphic. In this paper, we give a characterization for π to be potentially C _r ″-graphic. This work was supported by the grant of National Natural Science Foundation of China No. 10861006 and China Scholarship Council.     

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

Graphic sequences and split graphs

The split graph K _rVK _s on r+s vertices is denoted by S _r,s. A graphic sequence π = (d ₁, d ₂, ···, d _n) is said to be potentially S _r,s-graphic if there is a realization of π containing S _r,s as a subgraph. In this paper, a simple sufficient condition for π to be potentially S _r,s-graphic is obtained, which extends an analogous condition for p to be potentially K _r+1-graphic due to Yin and Li (Discrete Math. 301 (2005) 218–227). As an application of this condition, we further determine the values of σ(S _r,s, n) for n ≥ 3r + 3s - 1.     

The Erdös-Jacobson-Lehel conjecture on potentiallyP k-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d₁, d₂,⋯, d₂) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyP _k-graphic if it has a realizationG having propertyP _k . The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d₁, d₂,…, d₂) without zero terms and with degree sum σ(π)=(d ₁+d ₂+ …+d ₂) at least σ(k,n) is potentially P_k-graphic has been proved positive. Project supported by the National Natural Science Foundation of China (Grant No. 19671077) and the Doctoral Program Foundation of National Education Department of China.     

A Rao-type characterization for a sequence to have a realization containing a split graph

For a given graph H, a non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be potentially H-graphic if there exists a realization of π containing H as a subgraph. The split graph on r+s vertices is denoted by S_r,s. In this paper, we give a Rao-type characterization for π to be potentially S_r,s-graphic. A simplification of this characterization is also presented.     

On potentially <Emphasis Type="Italic">H</Emphasis>-graphic sequences

For given a graph H, a graphic sequence π = (d ₁, d ₂,..., d _n) is said to be potentially H-graphic if there is a realization of π containing H as a subgraph. In this paper, we characterize the potentially (K ₅ − e)-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence π to be potentially K ₅-graphic, where K _r is a complete graph on r vertices and K _r-e is a graph obtained from K _r by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence π to be potentially K ₆-graphic. Project supported by National Natural Science Foundation of China (No. 10401010).     

The Characterization for a Graphic Sequence to have a Realization Containing K 1,1,s

A graphic sequence π?=?(d ₁, d ₂, . . . , d _n ) is said to be potentially K _1,1,s -graphic if there is a realization of π containing K _1,1,s as a subgraph, where K _1,1,s is the 1?× 1?× s complete 3-partite graph. In this paper, a simple characterization of potentially K _1,1,s -graphic sequences for s?≥ 2 and n?≥ 3s?+?1 is obtained. This characterization implies Lai’s conjecture on σ(K _1,1,s , n), which was confirmed by J.H. Yin, J.S. Li and W.Y. Li, and the values of σ(K _2,s , n) for s?≥ 4 and n?≥ 3s?+?1, where K _2,s is the 2?× s complete bipartite graph.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

有一个实现包含Kr +1的可图序列

设K _{r +1}是一个r +1个顶点的完全图. 一个可图序列π =(d₁, d₂,…, d_n)称为是蕴含K _r+1 -可图的, 如果π有一个实现包含 K _{r +1}作为子图. 该文进一步研究了蕴含K _{r+1 -}可图序列的一些新的条件, 证明了这些条件包含文献[14,10,11]中的一些主要结果和当n≥5r/2 +1时,σ(K _r+1, n)之值(此值在文献[2]中被猜测, 在文献[6,7,8,3]中被证实). 此外, 确定了所有满足n≥5, d₅≥4 且不蕴含K₅ -可图序列π=(d₁, d₂,…, d_n)的集合.     

On the Davenport constant and on the structure of extremal zero-sum free sequences

Let $G = C_{n_1 } \oplus \cdots \oplus C_{n_r }$ with 1 < n ₁ | ?? | n _r be a finite abelian group, d*(G) = n ₁ +??+n _r ?r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) ?? d*(G), and the standing conjecture is that equality holds for G = C _n ^r . We show that equality does not hold for C ₂ ?? C _2n ^r , where n ?? 3 is odd and r ?? 4. This gives new information on the structure of extremal zero-sum free sequences over C _2n ^r .     

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.     

The action of a few permutations on r-tuples is quickly transitive

We prove that for every r and d≥2 there is a C such that for most choices of d permutations π₁, π₂,…,π_d of S_n, the following holds: for any two r-tuples of distinct elements in {1,…,n}, there is a product of less than C log n of the π_is which map the first r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to a general context of which random Cayley graph quotients of S_n are good expanders. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 335–350, 1998     

A generalization of a conjecture due to Erd?s, Jacobson and Lehel

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is r-graphic if it is realizable by an r-graph on n vertices. Let be the smallest even integer such that each n-term r-graphic sequence with term sum of at least is realizable by an r-graph containing as a subgraph. In this paper, we determine the value of for sufficiently large n, which generalizes a conjecture due to Erd?s, Jacobson and Lehel.     

The packing problem for projective geometries over GF(3) with dimension greater than five

An (n, d) set in the projective geometry PG(r, q) is a set of n points, no d of which are dependent. The packing problem is that of finding n(r, q, d), the largest size of an (n, d) set in PG(r, q). The packing problem for PG(r, 3) is considered. All of the values of n(r, 3, d) for r ? 5 are known. New results for r = 6 are n(6, 3, 5) = 14 and 20 ? n(6, 3, 4) ? 31. In general, upper bounds on n(r, q, d) are determined using a slightly improved sphere-packing bound, the linear programming approach of coding theory, and an orthogonal (n, d) set with the known extremal values of n(r, q, d)—values when r and d are close to each other. The BCH constructions and computer searches are used to give lower bounds. The current situation for the packing problem for PG(r, 3) with r ? 15 is summarized in a final table.     

Near automorphisms of cycles

Let f be a permutation of V(G). Define δ_f(x,y)=|d_G(x,y)-d_G(f(x),f(y))| and δ_f(G)=∑δ_f(x,y) over all the unordered pairs {x,y} of distinct vertices of G. Let π(G) denote the smallest positive value of δ_f(G) among all the permutations f of V(G). The permutation f with δ_f(G)=π(G) is called a near automorphism of G. In this paper, we study the near automorphisms of cycles C_n and we prove that π(C_n)=4⌊n/2⌋-4, moreover, we obtain the set of near automorphisms of C_n.