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).     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

1-Factorizations of cartesian products of regular graphs

Let G₁, G₂…, G_n be regular graphs and H be the Cartesian product of these graphs (H = G₁ × G₂ × … × G_n). The following will be proved: If the set {G₁, G₂…, G_n} has at leat one of the following properties: (*) for at leat one i ? {1, 2,…, n}, there exists a 1-factorization of G_i or (**) there exists at least two numbers i and j such that 1 ≤ i < j ≤ n and both the Graphs G_i and G_j contain at least one 1-factor, then there exists a 1-factorization of H. Further results: Let F be a cycle of length greater than three and let G be an arbitrary cubic graph. Then there exists a 1-factorization of the 5-regular graph H = F × G. The last result shows that neither (*) nor (**) is a necessary condition for the existence of a 1-factorization of a Cartesian product of regular graphs.     

Some results on starlike trees and sunlike graphs

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H ₁,H ₂, ...H _n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH _i with thei-th vertex ofG. In particular, ifH is the family of the paths $P_{k_1 } , P_{k_2 } , ..., P_{k_n } $ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k ₁,k ₂, …,k _n ). For any (x ₁,x ₂, …,x _n ) ∈I _* ⁿ , whereI _*={0,1}, letG(x ₁,x ₂, …,x _n ) be the subgraph ofG which is obtained by deleting the verticesi ₁, i₂, …,i _j ∈ V(G) (0≤j≤n), provided that $x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$ . LetG(x ₁,x ₂,…, x _n] be the characteristic polynomial ofG(x ₁,x ₂,…, x _n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that $$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$ where x=(x ₁,x ₂,…,x _n );G[k ₁,k ₂,…,k _n ] andP _n (γ) denote the characteristic polynomial ofG(k ₁,k ₂,…,k _n ) andP _n , respectively. Besides, ifG is a graph with λ₁(G)≥1 we show that λ₁(G)≤λ₁(G(k ₁,k ₂, ...,k _n )) < for all positive integersk ₁,k ₂,…,k _n , where λ₁ denotes the largest eigenvalue.     

On B(4, 13) 2-Groups

A group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j  | 1 ≤ i, j ≤ n}| ≤k. In this article, we give a complete characterization of B(4, 13) 2-groups, and then obtain a complete characterization of B(4, 13) groups.     

Convergence in homogeneous random graphs

For a sequence p? = (p(1), p(2), ‥ .), let G(n, p?) denote the random graph with vertex set {1, 2, ‥ ., n} in which two vertices i, j are adjacent with probability p(|i ? j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n, p?), can be affected by the behaviour of p? and the strength of the language we use.     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

Orientations of circle graphs

Let C(v₁, …,v_n) be a system consisting of a circle C with chords v₁, …,v_n on it having different endpoints. Define a graph G having vertex set V(G) = {v₁, …,v_n} and for which vertices v_i and v_j are adjacent in G if the chords v_i and v_j intersect. Such a graph will be called a circle graph. The chords divide the interior of C into a number of regions. We give a method which associates to each such region an orientation of the edges of G. For a given C(v₁, …,v_n) the number m of different orientations corresponding to it satisfies q + 1 ≤ m ≤ n + q + 1, where q is the number of edges in G. An oriented graph obtained from a diagram C(v₁, …,v_n) as above is called an oriented circle graph (OCG). We show that transitive orientations of permutation graphs are OCGs, and give a characterization of tournaments which are OCGs. When the region is a peripheral one, the orientation of G is acyclic. In this case we define a special orientation of the complement of G, and use this to develop an improved algorithm for finding a maximum independent set in G.     

An asymptotic version of a conjecture by Enomoto and Ota

In 2000, Enomoto and Ota [J Graph Theory 34 (2000), 163–169] stated the following conjecture. Let G be a graph of order n, and let n₁, n₂, …, n_k be positive integers with \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} {{n}}_{{{i}}} = {{n}}\end{eqnarray*}. If σ₂(G)≥n+ k?1, then for any k distinct vertices x₁, x₂, …, x_k in G, there exist vertex disjoint paths P₁, P₂, …, P_k such that |P_i|=n_i and x_i is an endpoint of P_i for every i, 1≤i≤k. We prove an asymptotic version of this conjecture in the following sense. For every k positive real numbers γ₁, …, γ_k with \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} \gamma_{{{i}}} = {{1}}\end{eqnarray*}, and for every ε>0, there exists n₀ such that for every graph G of order n≥n₀ with σ₂(G)≥n+ k?1, and for every choice of k vertices x₁, …, x_k∈V(G), there exist vertex disjoint paths P₁, …, P_k in G such that \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} |{{P}}_{{{i}}}| = {{n}}\end{eqnarray*}, the vertex x_i is an endpoint of the path P_i, and (γ_i?ε)n<|P_i|<(γ_i + ε)n for every i, 1≤i≤k. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 37–51, 2010     

Covering the vertices of a graph by cycles of prescribed length

The main theorem of that paper is the following: let G be a graph of order n, of size at least (n² - 3n + 6)/2. For any integers k, n₁, n₂,…,n_k such that n = n₁ + n₂ +. + n_k and n_i ? 3, there exists a covering of the vertices of G by disjoint cycles (C_i) =l…k with |C_i| = ni, except when n = 6, n₁ = 3, n₂ = 3, and G is isomorphic to G₁, the complement of G₁ consisting of a C₃ and a stable set of three vertices, or when n = 9, n₁ = n₂ = n₃ = 3, and G is isomorphic to G₂, the complement of G₂ consisting of a complete graph on four vertices and a stable set of five vertices. We prove an analogous theorem for bipartite graphs: let G be a bipartite balanced graph of order 2n, of size at least n² - n + 2. For any integers s, n₁, n₂,…,n_s with n_i ? 2 and n = n₁ + n₂ + ? + n_s, there exists a covering of the vertices of G by s disjoint cycles C_i, with |C_i| = 2n_i.     

Induced matching extendable graphs

We say that a simple graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. The main results of this paper are as follows: (1) For every connected IM-extendable graph G with |V(G)| ≥ 4, the girth g(G) ≤ 4. (2) If G is a connected IM-extendable graph, then |E(G)| ≥ ${3\over 2}|V(G)| - 2$; the equality holds if and only if G ≅ T × K₂, where T is a tree. (3) The only 3-regular connected IM-extendable graphs are C_n × K₂, for n ≥ 3, and C_2n(1, n), for n ≥ 2, where C_2n(1, n) is the graph with 2n vertices x₀, x₁, …, x_2n−1, such that x_ix_j is an edge of C_2n(1, n) if either |i − j| ≡ 1 (mod 2n) or |i − j| ≡ n (mod 2n). © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 203–213, 1998     

Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_{i = 1} a_i and σ₂(G) ≥ n + k − 1, then for any k vertices v₁, v₂,…, v_k in G, there exist vertex‐disjoint paths P₁, P₂,…, P_k such that |V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all a_i ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

OD-characterization of Almost Simple Groups Related to U3（5）

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.     

Mixed ramsey numbers: Chromatic numbers versus graphs

For positive integers n₁, n₂, …, n_I and graphs G_I+1, G_I+2, …, G_k, 1 ≤ / < k, the mixed Ramsey number _χ(n₁, …, n₁, G_I+1, …, G_k) is define as the least positive integer p such that for each factorization K_p = F₁⊕ … ⊕ F_⊕ F_I+1⊕ … ⊕ F_k, it it follows that χ(F_i) ≥ n_i for some i, 1 ? i ? l, or G_i is a subgraph of F_i for some i, l < i ? k. Formulas are presented for maxed Ramsey numbers in which the graphs G_I+1, G_I+2, …, G_k are connected, and in which k = I+1 and G_I+1 is arbitray.     

Double cycle covers and the petersen graph

Let O(G) denote the set of odd-degree vertices of a graph G. Let t ? N and let ??_t denote the family of graphs G whose edge set has a partition. E(g) = E₁ U E₂ U … U E_tsuch that O(G) = O(G[E_i]) (1 ? i ? t). This partition is associated with a double cycle cover of G. We show that if a graph G is at most 5 edges short of being 4-edge-connected, then exactly one of these holds: G ? ??₃, G has at least one cut-edge, or G is contractible to the Petersen graph. We also improve a sufficient condition of Jaeger for G ? ??_2p+1(p ? N).     

On 2-factors with cycles containing specified vertices in a bipartite graph

Let k≥1 be an integer and G=(V ₁,V ₂;E) a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+2. Our result is as follows: If $d(x)+d(y)\geq \lceil\frac{4n+k}{3}\rceil$ for any nonadjacent vertices x∈V ₁ and y∈V ₂, then for any k distinct vertices z ₁,…,z _k , G contains a 2-factor with k+1 cycles C ₁,…,C _k+1 such that z _i ∈V(C _i ) and l(C _i )=4 for each i∈{1,…,k}.     

A class of facet producing graphs for vertex packing polyhedra

We examine a family of graphs called webs. For integers n ? 2 and $\text{k, 1 ? k ?}\text{1}\text{2}\text{n}$, the web W(n, k) has vertices V_n = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?V_n (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement.