The circular dimension of a graph

A graph is a pair (V, I), V being the vertices and I being the relation of adjacency on V. Given a graph G, then a collection of functions {f_i}^m_n=1, each f_i mapping each vertex of V into anarc on a fixed circle, is said to define an m-arc intersection model for G if for all x,y ? V, xly ? (∨_i?m)(f_i(x)∩f_i(y)≠Ø). The circular dimension of a graph G is defined as the smallest integer m such that G has an m-arc intersection model. In this paper we establish that the maximum circular dimension of any complete partite graph having n vertices is the largest integer p such that 2^p+p?n+1.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Generalized n-tuple colorings of a graph: A counterexample to a conjecture of Brigham and Dutton

In their papers (Technical Report CS-TR 50, University of Central Florida, 1980; J. Combin. Theory Ser. B32 (1982), 90–94) Brigham and Dutton introduce the notion of (n : i)-chromatic numbers of a graph, a generalization of Stahl's nth chromatic numbers (J. Combin. Theory Ser. B20, (1976), 185–203). The (n : i)-chromatic number of a graph G, denoted by χ_nⁱ(G), is the smallest integer m such that each vertex of G can be colored with a set of n colors in {1, 2,…, m} in such a way that any two adjacent vertices have exactly i colors in common. Brigham and Dutton conjecture at the end of loc cit that for all integers n and i with 0 ≤ i ≤ n ? 1, and for every graph G, χ_nⁱ⁺¹(G) ≤ χ_nⁱ(G). We prove this conjecture in some special cases and disprove it in the general case.     

The Ramsey number for a cycle of length six versus a clique of order eight

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_m denote a cycle of length m and K_n a complete graph of order n. In this paper, it is shown that R(C₆,K₈)=36.     

Nombres de ramsey dans le cas orienté

Given two directed graphs G₁, G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any partition {U₁,U₂} of the arcs of the complete symmetric directed graph K¹_n, there exists an integer i such that the partial graph generated by U_i contains G_i as a subgraph. In this article, we determine R(P?_m,D?_n) and R(D?_m,D?_n) for some values of m and n, where P?_m denotes the directed path having m vertices and D?_m is obtained from P?_m by adding an arc from the initial vertex of P?_m to the terminal vertex.     

Incidence coloring of the squares of some graphs

The incidence chromatic number of G, denoted by χ_i(G), is the least number of colors such that G has an incidence coloring. In this paper, we determine the incidence chromatic number of the powers of paths, trees, which are min{n,2k+1}, and Δ(T²)+1, respectively. For the square of a Halin graph, we give an upper bound of its incidence chromatic number.     

Relation between frobenius and 2-frobenius groups with order components of finite groups

IfG is a finite group, we define its prime graph Г(G), as follows: its vertices are the primes dividing the order ofG and two verticesp, q are joined by an edge, if there is an element inG of orderpq. We denote the set of all the connected components of the graph Г(G) by T(G)= {π_i(G), fori = 1,2, …,t(G)}, where t(G) is the number of connected components of Г(G). We also denote by π(n) the set of all primes dividingn, wheren is a natural number. Then ¦G¦ can be expressed as a product of m₁, m₂, …, m_t(G), where m_i’s are positive integers with π(m_i) = π_i. Thesem _i ’s are called the order components ofG. LetOC(G) := {m ₁,m ₂, …,m _t (G)} be the set of order components ofG. In this paper we prove that, if G is a finite group andOC(G) =OC(M), where M is a finite simple group witht(M) ≥ 2, thenG is neither Frobenius nor 2-Frobenius.     

Uniformly optimal graphs in some classes of graphs with node failures

The uniformly optimal graph problem with node failures consists of finding the most reliable graph in the class Ω(n,m) of all graphs with n nodes and m edges in which nodes fail independently and edges never fail. The graph G is called uniformly optimal in Ω(n,m) if, for all node-failure probabilities q∈(0,1), the graph G is the most reliable graph in the class of graphs Ω(n,m). This paper proves that the multipartite graphs K(b,b+1,…,b+1,b+2) are uniformly optimal in their classes Ω((k+2)(b+1),(k²+3k+2)(b+1)²/2−1), where k is the number of partite sets of size (b+1), while for i>2, the multipartite graphs K(b,b+1,…,b+1,b+i) are not uniformly optimal in their classes Ω((k+2)b+k+i,(k+2)(k+1)b²/2+(k+1)(k+i)b+k(k+2i−1)/2).     

Comparing Zagreb indices for connected graphs

It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M₂(G)/m≥M₁(G)/n, where M₁ and M₂ are the first and second Zagreb indices. Hansen and Vuki?evi? proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m−n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M₂/(m−1)>M₁/n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.     

A New Construction for Cohen–Macaulay Graphs

Let G be a finite simple graph on a vertex set V(G) = {x ₁₁,…, x _n1}. Also let m ₁,…, m _n  ≥ 2 be integers and G ₁,…, G _n be connected simple graphs on the vertex sets V(G _i ) = {x _i1,…, x _{im i} }. In this article, we provide necessary and sufficient conditions on G ₁,…, G _n for which the graph obtained by attaching the G _i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs.     

Clique graphs of time graphs

A simple, finite graph G is called a time graph (equivalently, an indifference graph) if there is an injective real function f on the vertices v(G) such that v_iv_j ∈ e(G) for v_i ≠ v_j if and only if |f(v_i) ? f(v_j)| ≤ 1. A clique of a graph G is a maximal complete subgraph of G. The clique graph K(G) of a graph G is the intersection graph of the cliques of G. It will be shown that the clique graph of a time graph is a time graph, and that every time graph is the clique graph of some time graph. Denote the clique graph of a clique graph of G by K²(G), and inductively, denote K(K^m?1(G)) by K^m(G). Define the index indx(G) of a connected time graph G as the smallest integer n such that Kⁿ(G) is the trivial graph. It will be shown that the index of a time graph is equal to its diameter. Finally, bounds on the diameter of a time graph will be derived.     

Note on a new coloring number of a graph

The distance coloring number X_d(G) of a graph G is the minimum number n such that every vertex of G can be assigned a natural number m ≤ n and no two vertices at distance i are both assigned i. It is proved that for any natural number n there exists a graph G with X_d(G) = n.     

Two characterizations of interchange graphs of complete m-partite graphs

A graph G is m-partite if its points can be partitioned into m subsets V₁,…,V_m such that every line joins a point in V_i with a point in V_j, i ≠ j. A complete m-partite graph contains every line joining V_i with V_j. A complete graph K_p has every pair of its p points adjacent. The nth interchange graph I_n(G) of G is a graph whose points can be identified with the K_n+1's of G such that two points are adjacent whenever the corresponding K_n+1's have a K_n in common.Interchange graphs of complete 2-partite and 3-partite graphs have been characterized, but interchange graphs of complete m-partite graphs for m > 3 do not seem to have been investigated. The main result of this paper is two characterizations of interchange graphs of complete m-partite graphs for m ≥ 2.     

Undirected power graphs of semigroups

The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and a ^m =b or b ^m =a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or p ^m . Particular attention is given to the multiplicative semigroup ℤ _n and its subgroup U _n , where G(U _n ) is a major component of G(ℤ _n ). It is proved that G(U _n ) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(U _n ) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(U _n ) has no Hamiltonian cycle.     

Some results on the spectral radii of bicyclic graphs

A bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. Let Δ(G) and ρ(G) denote the maximum degree and the spectral radius of a graph G, respectively. Let B(n) be the set of bicyclic graphs on n vertices, and B(n,Δ)={G∈B(n)∣Δ(G)=Δ}. When Δ≥(n+3)/2 we characterize the graph which alone maximizes the spectral radius among all the graphs in B(n,Δ). It is also proved that for two graphs G₁ and G₂ in B(n), if Δ(G₁)>Δ(G₂) and Δ(G₁)≥⌈7n/9⌉+9, then ρ(G₁)>ρ(G₂).     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

A construction of Gray codes inducing complete graphs

A binary Gray code G(n) of length n, is a list of all 2ⁿn-bit codewords such that successive codewords differ in only one bit position. The sequence of bit positions where the single change occurs when going to the next codeword in G(n), denoted by S(n)?s₁,s₂,…,s_2ⁿ-1, is called the transition sequence of the Gray code G(n). The graph G_G(n) induced by a Gray code G(n) has vertex set {1,2,…,n} and edge set {{s_i,s_i+1}:1?i?2ⁿ-2}. If the first and the last codeword differ only in position s_2ⁿ, the code is cyclic and we extend the graph by two more edges {s_2ⁿ-1,s_2ⁿ} and {s_2ⁿ,s₁}. We solve a problem of Wilmer and Ernst [Graphs induced by Gray codes, Discrete Math. 257 (2002) 585-598] about a construction of an n-bit Gray code inducing the complete graph K_n. The technique used to solve this problem is based on a Gray code construction due to Bakos [A. Ádám, Truth Functions and the Problem of their Realization by Two-Terminal Graphs, Akadémiai Kiadó, Budapest, 1968], and which is presented in D.E. Knuth [The Art of Computer Programming, vol. 4, Addison-Wesley as part of “fascicle” 2, USA, 2005].     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.