Nonadjacent Vertices Research Articles

Paths and trails in edge-colored weighted graphs

Let (G,c,w) be an edge-colored weighted graph, where G is a nontrivial connected graph, c is an edge-coloring of G, and w is an edge-weighting of G. A path, a trail, a cycle, or a closed trail of G, say F, is called proper under the edge-coloring c if every two consecutive edges of F receive different colors in c. Let s and t be two specified nonadjacent vertices in G. In this paper, we study the problems for finding, in (G,c,w), the minimum weighted proper s-t-path, the minimum weighted proper s-t-trail, the minimum weighted proper cycle, the minimum weighted proper closed trail, the maximum weighted proper s-t-path, and the maximum weighted proper s-t-trail. When the minimization problems are considered we assume that (G,c,w) has no negative proper cycle, and when the maximization problems are considered we assume that (G,c,w) has no proper closed trail. We show that all these problems are solvable in polynomial time.

A Hamilton sufficient condition for completely independent spanning tree

Let T1,T2,…,Tk be spanning trees of a graph G. For any two vertices u,v of G, if the paths from u to v in these k trees are pairwise openly disjoint, then we say that T1,T2,…,Tk are completely independent. It has been asked whether sufficient conditions for Hamiltonian graphs are also sufficient for the existence of two completely independent spanning trees (CISTs). In this paper, we prove that it is true for aHamilton-condition. That is, if G is a graph with n vertices and |N(x)∪N(y)|≥n2, |N(x)∩N(y)|≥3 for every two nonadjacent vertices x,y of G, then G has two completely independent spanning trees.

Spanning k-tree with specified vertices

A [Formula: see text]-tree is a tree with maximum degree at most [Formula: see text]. For a graph [Formula: see text] and [Formula: see text] with [Formula: see text], let [Formula: see text] be the cardinality of a maximum independent set containing [Formula: see text] and [Formula: see text]. For a graph [Formula: see text] and [Formula: see text], the local connectivity [Formula: see text] is defined to be the maximum number of internally disjoint paths connecting [Formula: see text] and [Formula: see text] in [Formula: see text]. In this paper, we prove the following theorem and show the condition is sharp. Let [Formula: see text], [Formula: see text] and [Formula: see text] be integers with [Formula: see text], [Formula: see text] and [Formula: see text]. For any two nonadjacent vertices [Formula: see text] and [Formula: see text] of [Formula: see text], we have [Formula: see text] and [Formula: see text]. Then for any [Formula: see text] distinct vertices of [Formula: see text], [Formula: see text] has a spanning [Formula: see text]-tree such that each of [Formula: see text] specified vertices has degree at most [Formula: see text]. This theorem implies H. Matsuda and H. Matsumura’s result in [on a [Formula: see text]-tree containing specified cleares in a graph, Graphs Combin. 22 (2006) 371–381] and V. Neumann-Lara and E. Rivera-Campo’s result in [Spanning trees with bounded degrees, Combinatorica 11 (1991) 55–61].

Linial's Conjecture for Arc-spine Digraphs

A path partitionP of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition P of D is defined as ∑P∈Pmin⁡{|Pi|,k}. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by πk(D). A stable set of a digraph D is a subset of pairwise non-adjacent vertices of V(D). Given a positive integer k, we denote by αk(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that πk(D) ≤ αk(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial's Conjecture for arc-spine digraphs.

Partially Commutative Metabelian Pro-P-Groups

We prove two theorems about a partially commutative metabelian pro-p-group. The first theorem concerns the structure of annihilators for the commutators of nonadjacent vertices of the defining graph, and the second discusses the centralizers of the vertices of the defining graph.

Eccentric connectivity coindex under graph operations

The eccentric connectivity coindex is defined as the total eccentricity sum of all non-adjacent vertex pairs in a connected graph. In this paper, we present exact expressions or sharp lower bounds for the eccentric connectivity coindex of several graph operations such as sum, disjunction, symmetric difference, lexicographic product, generalized hierarchical product, Cartesian product, rooted product, corona product, and strong product.

On the Manoussakis Conjecture for a Digraph to be Hamiltonian

Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. Conjecture. Let D be a 2-strongly connected digraph of order n such that for all distinct pairs of non-adjacent vertices x, y and w, z, we have d(x)+d(y)+d(w)+d(z) ≥ 4n − 3. Then D is Hamiltonian. In this note, we prove that if D satisfies the conditions of this conjecture, then (i) D has a cycle factor; (ii) If {x, y} is a pair of non-adjacent vertices of D such that d(x) + d(y) ≤ 2n − 2, then D is Hamiltonian if and only if D contains a cycle passing through x and y; (iii) If {x, y} a pair of non-adjacent vertices of D such that d(x)+d(y) ≤ 2n−4, then D contains cycles of all lengths 3, 4, . . . , n−1; (iv) D contains a cycle of length at least n − 1

Topological bounds on the dimension of orthogonal representations of graphs

An orthogonal representation of a graph is an assignment of nonzero real vectors to its vertices such that distinct non-adjacent vertices are assigned to orthogonal vectors. We prove general lower bounds on the dimension of orthogonal representations of graphs using the Borsuk–Ulam theorem from algebraic topology. Our bounds strengthen the Kneser conjecture, proved by Lovász in 1978, and some of its extensions due to Bárány, Schrijver, Dol’nikov, and Kriz. As applications, we determine the integrality gap of fractional upper bounds on the Shannon capacity of graphs and the quantum one-round communication complexity of certain promise equality problems.

On Traceability of Claw-o−1-heavy Graphs

A graph is called traceable if it contains a Hamilton path, i.e., a path passing through all its vertices. Let $G$ be a graph on $n$ vertices. $G$ is called claw-$o_{-1}$-heavy if every induced claw ($K_{1,3}$) of $G$ has a pair of nonadjacent vertices with degree sum at least $n-1$ in $G$. In this paper we show that a claw-$o_{-1}$-heavy graph $G$ is traceable if we impose certain additional conditions on $G$ involving forbidden induced subgraphs.

On bicliques and the second clique graph of suspensions

The clique graphK(G) of a graph G is the intersection graph of the set of all (maximal) cliques of G. The second clique graphK2(G)ofG is defined as K2(G)=K(K(G)). The main motivation for this work is to attempt to characterize the graphs G that maximize |K2(G)|, as has been done for |K(G)| by Moon and Moser in 1965.The suspensionS(G) of a graph G is the graph that results from adding two non-adjacent vertices to the graph G, that are adjacent to every vertex of G. Using a new biclique operator B that transforms a graph G into its biclique graphB(G), we found the characterization K2(S(G))≅B(K(G)). We also found a characterization of the graphs G, that maximize |B(G)|.Here, a biclique (X,Y) of G is an ordered pair of subsets of vertices of G (not necessarily disjoint), such that every vertex x∈X is adjacent or equal to every vertex y∈Y, and such that (X,Y) is maximal under component-wise inclusion. The biclique graph B(G) of the graph G, is the graph whose vertices are the bicliques of G and two vertices (X,Y) and (X′,Y′) are adjacent, if and only if X∩X′≠0̸ or Y∩Y′≠0̸.

Necessary and sufficient condition for contextuality from incompatibility

Measurement incompatibility is the most basic resource that distinguishes quantum from classical physics. Contextuality is the critical resource behind the power of some models of quantum computation and is also a necessary ingredient for many applications in quantum information. A fundamental problem is thus identifying when incompatibility produces contextuality. Here, we show that, given a structure of incompatibility characterized by a graph in which nonadjacent vertices represent incompatible ideal measurements, the necessary and sufficient condition for the existence of a quantum realization producing contextuality is that this graph contains induced cycles of size larger than three.

Partite Saturation of Complete Graphs

We study the problem of determining sat$(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two nonadjacent vertices from different parts creates a $K_r$. Improving recent results of Ferrara, Jacobson, Pfender, and Wenger, and generalizing a recent result of Roberts, we show that sat$(n,k,r) = \alpha(k,r)n + O(1)$ as $n \rightarrow \infty$. Moreover, for every $3 \leq r \leq k$ we prove that $k(2k - 4) \leq \alpha(k,r) \leq (k-1)(4 r - \ell -6),$ where $\ell = \mbox{min}\{k, 2r-3\}$, and show that the lower bound is tight for infinitely many values of $r$ and every $k\geq 2r-1$. This allows us to prove that, for these values, sat$(n,k,r) = k(2r-4)n + O(1)$ as $n \rightarrow \infty$. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.

On Critical 3-Connected Graphs with Two Vertices of Degree 3. Part I

A 3-connected graph G is said to be critical if for any vertex υ ∈ V (G) the graph G − υ is not 3-connected. Entringer and Slater proved that any critical 3-connected graph contains at least two vertices of degree 3. In this paper, a classification of critical 3-connected graphs with two vertices of degree 3 is given in the case where these vertices are adjacent. The case of nonadjacent vertices of degree 3 will be studied in the second part of the paper, which will be published later.

Toll number of the strong product of graphs

A tolled walk T between two non-adjacent vertices u and v in a graph G is a walk, in which u is adjacent only to the second vertex of T and v is adjacent only to the second-to-last vertex of T. A toll interval between u,v∈V(G) is a set TG(u,v)={x∈V(G)∣xlies on a tolled walk betweenuandv}. A set S⊆V(G) is toll convex, if TG(u,v)⊆S for all u,v∈S. A toll closure of a set S⊆V(G) is the union of toll intervals between all pairs of vertices from S. The size of a smallest set S whose toll closure is the whole vertex set is called the toll number of G, tn(G). This paper investigates the toll number of the strong product of graphs. First, a description of toll intervals between two vertices in the strong product graphs is given. Using this result we characterize graphs with tn(G⊠H)=2 and graphs with tn(G⊠H)=3, which are the only two possibilities. As an addition, for the t-hull number of G⊠H we show that th(G⊠H)=2 for any non-complete graphs G and H. As extreme vertices play an important role in different convexity types, we show that no vertex of the strong product graph of two non-complete graphs is an extreme vertex with respect to the toll convexity.

Edge clique cover of claw‐free graphs

AbstractThe smallest number of cliques, covering all edges of a graph , is called the (edge) clique cover number of and is denoted by . It is an easy observation that if is a line graph on vertices, then . G. Chen et al. [Discrete Math. 219 (2000), no. 1–3, 17–26; MR1761707] extended this observation to all quasi‐line graphs and questioned if the same assertion holds for all claw‐free graphs. In this paper, using the celebrated structure theorem of claw‐free graphs due to Chudnovsky and Seymour, we give an affirmative answer to this question for all claw‐free graphs with independence number at least three. In particular, we prove that if is a connected claw‐free graph on vertices with three pairwise nonadjacent vertices, then and the equality holds if and only if is either the graph of icosahedron, or the complement of a graph on vertices called “twister” or the power of the cycle , for some positive integer .

Global rigidity of triangulations with braces

A.L. Cauchy proved that if the vertex-edge graphs of two convex polyhedra are isomorphic and corresponding faces are congruent then the two polyhedra are the same. This result implies that a convex polyhedron with triangular faces, as a bar-and-joint framework, is rigid.A framework is said to be globally rigid if the bar lengths uniquely determine all pairwise distances between the joints. Global rigidity implies rigidity. It is well-known that every three-dimensional generic bar-and-joint realisation of the graph of a convex polyhedron with triangular faces (that we call a triangulation) is rigid. It is also known that if the number of vertices is at least five then such a realisation of a triangulation is never globally rigid.In this paper we consider braced triangulations, obtained from triangulations by adding a set of extra bars (bracing edges) connecting pairs of non-adjacent vertices. We show that a braced triangulation is generically globally rigid in three-space if and only if it is 4-connected. The special case, when there is only one bracing edge, verifies a conjecture of W. Whiteley.Our proof is based on a new result on the vertex splitting operation which may be of independent interest. We show that every graph which can be obtained from the complete graph on five vertices by non-trivial vertex splitting operations is generically globally rigid in three-space.

A degree condition for fractional [a,b]-covered graphs

Let G be a graph of order n with δ(G)≥a+1, and 3≤a≤b be integers. In this paper, we first show that if G satisfies max⁡{dG(x),dG(y)}≥a(n+1)a+b for each pair of nonadjacent vertices x,y of G, then G is a fractional [a,b]-covered graph. It is a generalization of the known result with a=b=k which is given by Zhou. Furthermore, we show that this result is best possible in some sense.

A fast deterministic detection of small pattern graphs in graphs without large cliques

We show that for several pattern graphs on four vertices (e.g., C4), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n2.5719k0.3176+n2k2) time for k<n0.394 and O(n2.5k0.5+n2k2) time for k≥n0.394. The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., K4), in host graphs with n vertices and no independent set on k+1 vertices.By using the concept of Ramsey numbers, we can extend our method for induced subgraph isomorphism to include larger pattern graphs having a set of independent vertices with the same neighborhood and n-vertex host graphs without cliques on k+1 vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).

Graphs with Conflict-Free Connection Number Two

An edge-colored graph G is conflict-free connected if every two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is the smallest number of colors needed in order to make G conflict-free connected. For a graph G, let C(G) be the subgraph of G induced by its set of cut-edges. In this paper, we first show that, if G is a connected non-complete graph G of order $$n\ge 9$$ with C(G) being a linear forest and with the minimum degree $$\delta (G)\ge \max \{3, \frac{n-4}{5}\}$$ , then $$cfc(G)=2$$ . The bound on the minimum degree is best possible. Next, we prove that, if G is a connected non-complete graph of order $$n\ge 33$$ with C(G) being a linear forest and with $$d(x)+d(y)\ge \frac{2n-9}{5}$$ for each pair of two nonadjacent vertices x, y of V(G), then $$cfc(G)=2$$ . Both bounds, on the order n and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on G and the number of cut edges in G.

Extending Vertex and Edge Pancyclic Graphs

A graph G of order $$n\ge 3$$ is pancyclic if G contains a cycle of each possible length from 3 to n, and vertex pancyclic (edge pancyclic) if every vertex (edge) is contained on a cycle of each possible length from 3 to n. A chord of a cycle is an edge between two nonadjacent vertices of the cycle, and chorded cycle is a cycle containing at least one chord. We define a graph G of order $$n\ge 4$$ to be chorded pancyclic if G contains a chorded cycle of each possible length from 4 to n. In this article, we consider extensions of the property of being chorded pancyclic to chorded vertex pancyclic and chorded edge pancyclic.