On k-ordered Graphs Involved Degree Sum

Abstract A graph G is k-ordered Hamiltonian,2≤k≤n,if for every ordered sequence S of k distinctvertlces of G,there exists a Hamiltonian cycle that encounters S in the given order. In this article, we provethat if G is a graph on n vertices with degree sum of nonadjacent vertices at least n+3k-9/2,then G is k-orderedHamiltonian for k=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n+2[k/2]-2 ifk(G)≥3k-1/2 or δ(G)≥5k-4.Several known results are generalized.     

On the Edge-forwarding Indices of Frobenius Graphs

A G-Frobenius graph F, as defined by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K × H with Frobenius kernel K and Frobenius complement H. F is also shown to be a Cayley graph, F = Cay（K, S） for K and some subset S of the group K. On the other hand, a network N with a routing function R, written as （N, R）, is an undirected graph N together with a routing R which consists of a collection of simple paths connecting every pair of vertices in the graph. The edge-forwarding index π（N） of a network （N, R）, defined by Heydemann, Meyer, and Sotteau, is a parameter to describe tile maximum load of edges of N. In this paper, we study the edge-forwarding indices of Frobenius graphs. In particular, we obtain the edge-forwarding index of a G-Frobenius graph F with rank（G） ≤ 50.     

On Functors of Finite Degree and κ-Metrizable Bicompact Spaces

Siberian Mathematical Journal -     

On （g, f）-Uniform Graphs

A graph G is called a (g, f)-uniform graph if for each edge of G, there is a (g, f)-factor containing it and another (g, f)-factor excluding it. In this paper a necessary and sufficient condition for a graph to be a (g, f)-uniform graph is given and some applications of this condition are discussed. In particular, some simple sufficient conditions for a graph to be an [a, b]-uniform graph are obtained for a≤b.     

AVDTC Numbers of Generalized Halin Graphs with Maximum Degree at Least 6

In a paper by Zhang and Chen et al.（see [11]）, a conjecture was made concerning the minimum number of colors Xat（G） required in a proper total-coloring of G so that any two adjacent vertices have different color sets, where the color set of a vertex v is the set composed of the color of v and the colors incident to v. We find the exact values of Xat（G） and thus verify the conjecture when G is a Generalized Halin graph with maximum degree at least 6, A generalized Halin graph is a 2-connected plane graph G such that removing all the edges of the boundary of the exterior face of G （the degrees of the vertices in the boundary of exterior face of G are all three） gives a tree.     

On the Degree of Approximation by Spherical Translations

The degree of approximation of spherical functions by the translations formed by a function defined on the unit sphere is dealt with. A kind of Jackson inequality is established under the condition that none of the L^2（S^q） norms of the orthogonal projection operators of the translated function are zeros. In the present paper we show that the spherical translations share the same degree of approximation as that of spherical harmonics.     

On the Gracefulness of Two Family of the Hexagonal Graphs

Thegracefulgraphswaspresentedinl966.Sincethattime,manyresultshavebeenobtainedongacefulgraphsandit'smanyapplicationsappear(see[2j,L4j).Thegraceful-nessofP.XP.andC.XP,areprovedbythepaper[1],[3j,[5j,[6]-Followingthis,inthispaper,itisobtainedthatthegracefu1nessofthefamiliesofthehexagonalgraphsG,(n)andG2(n).Fornotationsandteminologiesonthegracelulgraphnotdefinedhere,wefollowthereference[3j-NowletG(V,E)beagraph,6bethelabelingoftheverticesofG,andforanyedgeuv,define6(uv)=lo(u)-o(v)Iitistheedge's…     

无爪图的次与Hamilton连通性

本文证明了：设G是n阶、k（≥3）连通无爪图，且不含同构于B的导出子图，若存在点v_0∈V（G），使d（v_0）≥n-2k＋4，则G是Hamilton连通的．     

Mutlhews-Sumner定理的证明

本文证明了：若Ｇ是２连通无爪图且δ（Ｇ）≥ｎ－２３,则Ｇ是Ｈａｍｉｌｔｏｎ图     

On（g,f）—Uniform Graphs

Thegraphsconsideredinthispaperwillbesimpleundirectedgraphs.LetGbeagraphwithvertexsetV(G)andedgesetE(G).ForavertexxofG,thedegreeofxinGisdenotedbydG(x).Theminimumdegreeandthemaximumdegree0fGaredenotedbyS(G)andb(G),respectively.Letgandfbetw0integer-valuedfunctionsdefined0nV(G)suchthatg(x)5f(x)foreveryx6V(G).Thena(g,f)-factorofGisaspanningsubgraphFofGsatisfyingg(x)SdF(x)5f(x)forallxEV(G).Ifg(x)=f(x)foreachxEV(G),thena(g,f)-factoriscalledanf-factor.Iffisaconstantfunctiontakingthevaluek,…     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

Degree sequences of <Emphasis Type="Italic">k</Emphasis>-multi-hypertournaments

Let n and k（n ≥ k 〉 1） be two non-negative integers.A k-multi-hypertournament on n vertices is a pair（V,A）,where V is a set of vertices with ｜V｜=n,and A is a set of k-tuples of vertices,called arcs,such that for any k-subset S of V,A contains at least one（at most k!） of the k! k-tuples whose entries belong to S.The necessary and suffcient conditions for a non-decreasing sequence of non-negative integers to be the out-degree sequence（in-degree sequence） of some k-multi-hypertournament are given.     

On the <Emphasis Type="Italic">k</Emphasis>-gamma <Emphasis Type="Italic">q</Emphasis>-distribution

We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.     

Generalizing Pancyclic and <Emphasis Type="Italic">k</Emphasis>-Ordered Graphs

Given positive integers kmn, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with mrn, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be (k,m)-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is (k,m)-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.Acknowledgments. The authors would like to thank the referees for their careful reading of the paper and their useful suggestions.Final version received: January 26, 2004     

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

Optimal (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) visual cryptographic schemes for general <Emphasis Type="Italic">k</Emphasis>

In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L ₁-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.     

Sparse <Emphasis Type="Italic">H</Emphasis>-Colourable Graphs of Bounded Maximum Degree

Let F be a graph of order at most k. We prove that for any integer g there is a graph G of girth at least g and of maximum degree at most 5k¹³ such that G admits a surjective homomorphism c to F, and moreover, for any F-pointed graph H with at most k vertices, and for any homomorphism h from G to H there is a unique homomorphism f from F to H such that h=fc. As a consequence, we prove that if H is a projective graph of order k, then for any finite family of prescribed mappings from a set X to V(H) (with ||=t), there is a graph G of arbitrary large girth and of maximum degree at most 5k^26mt (where m=|X|) such that and up to an automorphism of H, there are exactly t homomorphisms from G to H, each of which is an extension of an f.Supported in part by the National Science Council under grant NSC89-2115-M-110-012Final version received: June 9, 2003     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

On the family of diophantine triples {<Emphasis Type="Italic">k</Emphasis> + 1, 4<Emphasis Type="Italic">k</Emphasis>, 9<Emphasis Type="Italic">k</Emphasis> + 3}

We prove that if k is a positive integer and d is a positive integer such that the product of any two distinct elements of the set {k + 1, 4k, 9k + 3, d} increased by 1 is a perfect square, then d = 144k ³ + 192k ² + 76k + 8.      

Modular <Emphasis Type="Italic">W</Emphasis> and <Emphasis Type="Italic">H</Emphasis> Type Lie Algebras

The modular Witt algebra W(p, n) and H(p, 2n) are defined on the polynomial rings Z_p[x₁,...,x_n] and Z_p[X₁,...,x_n, y₁,...,y_n] respectively. We generalize Z_p[x₁,...,xn] and Zp[x₁,...,xn, y₁,...,y_n], so we get the generalized W-type and H-type modular Lie algebras. We find all the derivations of W(p, 1).AMS Subject Classification: Primary 17B40, 17B56.