Almost resolvable Pk-decompositions of complete graphs

An almost P_k-factor of G is a P_k-factor of G - { v } for some vertex v. An almost resolvable P_k-decomposition of λK_n is a partition of the edges of λK_n into almost P_k-factors. We prove that necessary and sufficient conditions for the existence of an almost resolvable P_k-decomposition of λK_n are n ≡ 1 (mod k) and λnk/2 ≡ 0 (mod k ?1).     

On 4-semiregular 1-factorizations of complete graphs and complete bipartite graphs

A 4-semiregular 1-factorization is a 1-factorization in which every pair of distinct 1-factors forms a union of 4-cycles. LetK be the complete graphK _2nor the complete bipartite graphK _{n, n} .We prove that there is a 4-semiregular 1-factorization ofK if and only ifn is a power of 2 andn2, and 4-semiregular 1-factorizations ofK are isomorphic, and then we determine the symmetry groups. They are known for the case of the complete graphK _2n ,however, we prove them in a different method.     

On defining numbers of circular complete graphs

Let d(σ) stand for the defining number of the colouring σ. In this paper we consider and for the onto χ-colourings γ of the circular complete graph K_n,d. In this regard we obtain a lower bound for d^min(K_n,d) and we also prove that this parameter is asymptotically equal to χ-1. Also, we show that when χ?4 and s≠0 then d^max(K_χd-s,d)=χ+2s-3, and, moreover, we prove an inequality relating this parameter to the circular chromatic number for any graph G.     

On complete intersection toric ideals of graphs

We study the graphs G for which their toric ideals I _G are complete intersections. In particular, we prove that for a connected graph G such that I _G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal.     

On the Ramsey multiplicity of complete graphs

We show that, for n large, there must exist at least $$\frac{{n^t }} {{C^{(1 + o(1)t^2 } )}}$$ monochromatic K _t s in any two-colouring of the edges of K _n , where C??2.18 is an explicitly defined constant. The old lower bound, due to Erd?s [2], and based upon the standard bounds for Ramsey??s theorem, is $$\frac{{n^t }} {{4^{(1 + o(1)t^2 } )}}. $$      

On the flow numbers of signed complete and complete bipartite graphs

We determine the flow numbers of signed complete and signed complete bipartite graphs.     

On complete graphs with negative r-mean curvature

We generalize Efimov's Theorem for graphs in Euclidean space using the scalar curvature, with an additional hypothesis on the second fundamental form.     

On the seidel integral complete multipartite graphs

For a simple undirected graph G, denote by A(G) the (0,1)-adjacency matrix of G. Let thematrix S(G) = J-I-2A(G) be its Seidel matrix, and let S _G (??) = det(??I-S(G)) be its Seidel characteristic polynomial, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. If all eigenvalues of S _G (??) are integral, then the graph G is called S-integral. In this paper, our main goal is to investigate the eigenvalues of S _G (??) for the complete multipartite graphs G = $G = K_{n_1 ,n_2 ,...n_t } $ . A necessary and sufficient condition for the complete tripartite graphs K _m,n,t and the complete multipartite graphs to be S-integral is given, respectively.     

On covering graphs by complete bipartite subgraphs

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m²/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.     

On balanced claw designs of complete multi-partite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of a balanced claw design BCD(m, n, c, λ) of a complete m-partite graph λK_m(n, n,…,n) is λ(m - 1)n ≡ 0 (mod 2c) and (m - 1)n ? c.     

On the complete chromatic number of Halin graphs

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.Write.1.IntroductionDefinition1.FOrany3-connectedplanargraphG(V,E,F)withA(G)23,iftheboundaryedgesoffacefowhichisadjacenttotheothersareremoved,itbecomesatree,andthedegreeofeachvertexofV(fo)is3,andthenGiscalledaHalingraph;foiscalledtheouterfaceofG,andtheotherscalledtheinteriorfaces,thevenicesonthefacefoarecalledtheoutervenices,theoillersarecalledtheinterior...ti..,tll.ForanyplanargraphG(V,E,F),f,f'eF,fisadjacenttof'ifan…     

On diagonal resolvable spheres

In this paper we give the complete answer to a question posed by A. Arhangel?skii and prove that the sphere Sⁿ$S^n$ is diagonal resolvable if and only if Sⁿ$S^n$ is an H  -space if and only if n∈{0,1,3,7}$n\in \{0,1,3,7\}$. Moreover, we prove that any upper half even dimensional Q$\mathbb{Q}$-sphere cannot be diagonal resolvable.     

Lattice complete graphs

We study the properties of graphs that can be placed in a rectangular lattice so that all vertices located in the same (horizontal or vertical) row be adjacent. Some criterion is formulated for an arbitrary graph to be in the specified class.     

Total edge irregularity strength of complete graphs and complete bipartite graphs

A total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G.We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs.     

Degree complete graphs

Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘?’ on the nonnegative integral vectors. It is clear that if S=(s₁,s₂,…,s_n) is an out-degree vector of an orientation of a graph G with vertices 1,2,…,n, then

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