Independence numbers of random subgraphs of distance graphs

We consider the distance graph G(n, r, s), whose vertices can be identified with r-element subsets of the set {1, 2,..., n}, two arbitrary vertices being joined by an edge if and only if the cardinality of the intersection of the corresponding subsets is s. For s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s–Ko–Rado problem and also play an important role in combinatorial geometry and coding theory. We study some properties of random subgraphs of G(n, r, s) in the Erd?s–Rényi model, in which every edge occurs in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, r, s) for the case of constant r and s. The independence number of a random subgraph is Θ(log₂n) times as large as that of the graph G(n, r, s) itself for r ≤ 2s + 1, while for r > 2s + 1 one has asymptotic stability: the two independence numbers asymptotically coincide.     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.     

Degree sum of a pair of independent edges and <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n ≥ 4 vertices such that G ∈ F if and only if d(e) + d(e’) ≥ 2n for every pair of independent edges e, e’ of G. We prove in this paper that for each G ∈ F, G is not Z ₃-connected if and only if G is one of K _2,n?2, K _3,n?3, K _2,n?2 ⁺ , K _3,n?3 ⁺ or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].     

An implicit degree condition for relative length of long paths and cycles in graphs

For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n + 1, then either G contains a hamiltonian path, or c(G) ≥ p(G) ? 1.     

On the extra edge-connectivity of hypercubes

The classical hypercube structure is a popular topological architecture in parallel computing environments and a large number of variations based on the hypercube were posed in the past three decades. Reliability evaluation of systems is important to the design and maintenance of multiprocessor systems. The h-extra edge-connectivity of graph G(V, E) is a kind of measure for the reliability of interconnection systems, which is defined as the minimum cardinality of a subset of edge set, if any, whose deletion disconnects G and such that every remaining component has at least h vertices. This paper shows that the h-extra edge-connectivity of the hypercube Qn is a constant 2~(n-1) for2~(n-1)/3 h ≤ 2~(n-1), and n ≥ 4, which extends the result of ["Bounding the size of the subgraph induced by m vertices and extra edge-connectivity of hypercubes, Discrete Applied Mathematics, 2013, 161(16): 2753-2757"].     

A note on the Engelking-Karłowicz theorem

We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Kar?owicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some a ∈ X ∩ Y the order-type of a ∩ X is different from that of a ∩ Y.In addition to the chromatic number x(G) of these graphs we study χ _κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.     

Minimum implicit degree condition restricted to claws for hamiltonian cycles

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G to be implicit 1-heavy (implicit 2-heavy) if at least one (two) of the end vertices of each induced claw has (have) implicit degree at least n/2. In this paper, we prove that: (a) Let G be a 2-connected graph of order n ≥ 3. If G is implicit 2-heavy and |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. (b) Let G be a 3-connected graph of order n ≥ 3. If G is implicit 1-heavy and |N(u) ∩ N(v)| ≥ 2 for each pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian.     

On one test for the switching separability of graphs modulo <Emphasis Type="Italic">q</Emphasis>

We consider graphs whose edges are marked by numbers (weights) from 1 to q - 1 (with zero corresponding to the absence of an edge). A graph is additive if its vertices can be marked so that, for every two nonadjacent vertices, the sum of the marks modulo q is zero, and for adjacent vertices, it equals the weight of the corresponding edge. A switching of a given graph is its sum modulo q with some additive graph on the same set of vertices. A graph on n vertices is switching separable if some of its switchings has no connected components of size greater than n - 2. We consider the following separability test: If removing any vertex from G leads to a switching separable graph then G is switching separable. We prove this test for q odd and characterize the set of exclusions for q even. Connection is established between the switching separability of a graph and the reducibility of the n-ary quasigroup constructed from the graph.     

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.     

Strictly Deza line graphs

For a given graph G, its line graph L(G) is defined as the graph with vertex set equal to the edge set of G in which two vertices are adjacent if and only if the corresponding edges of G have exactly one common vertex. A k-regular graph of diameter 2 on υ vertices is called a strictly Deza graph with parameters (υ, k, b, a) if it is not strongly regular and any two vertices have a or b common neighbors. We give a classification of strictly Deza line graphs.     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

Outer-2-independent domination in graphs

We initiate the study of outer-2-independent domination in graphs. An outer-2-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)?D has a neighbor in D and the maximum vertex degree of the subgraph induced by V(G)?D is at most one. The outer-2-independent domination number of a graph G is the minimum cardinality of an outer-2-independent dominating set of G. We show that if a graph has minimum degree at least two, then its outer-2-independent domination number equals the number of vertices minus the 2-independence number. Then we investigate the outer-2-independent domination in graphs with minimum degree one. We also prove the Vizing-type conjecture for outer-2-independent domination and disprove the Vizing-type conjecture for outer-connected domination.     

A note on the cubical dimension of new classes of binary trees

The cubical dimension of a graph G is the smallest dimension of a hypercube into which G is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with 2 ⁿ vertices, n ? 1, is n. The 2-rooted complete binary tree of depth n is obtained from two copies of the complete binary tree of depth n by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.     

Disjoint cliques in claw-free graphs

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k?s) and d(x)+d(y) ≥ n?2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K₃s and k ? s disjoint K₄s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.     

Graphs with small diameter determined by their <Emphasis Type="Italic">D</Emphasis>-spectra

Let G be a connected graph with vertex set V(G) = {v₁, v₂,..., v _n }. The distance matrix D(G) = (d _ij )_n×n is the matrix indexed by the vertices of G, where d _ij denotes the distance between the vertices v _i and v _j . Suppose that λ₁(D) ≥ λ₂(D) ≥... ≥ λ _n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.     

Cycles in dense digraphs

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ? E(G) such that G?X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?

We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ ½γ(G) (this would be best possible if true), and prove this conjecture in two special cases:

• when V(G) is the union of two cliques
• when the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

     

Phase transition for SIR model with random transition rates on complete graphs

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C _n with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d. copies of a positive random variable ρ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists λ _c > 0 such that when λ < λ _c ; the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → +∞ while when λ > λ _c ; there exist c(λ) > 0 and b(λ) > 0 such that for each n ≥ 1 with probability p ≥ b(λ); the proportion r_∞ ≥ c(λ): Furthermore, we prove that λ _c is the inverse of the production of the mean of ρ and the mean of the inverse of ξ.     

On the tree structure of the power digraphs modulo n

For any two positive integers n and k ? 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n ? 1} and such that there is a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P ₁, P ₂ ? P be such that P ₁ ∪ P ₂ = P and P ₁ ∩ P ₂ = ?. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P ₁. L. Somer and M. K?i?ek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

Graphs of nonsolvable groups with four degree-vertices

Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.