Vertex pancyclicity and new sufficient conditions

A lexicographic product of graphs G and H, denoted by is defined as a graph with the vertex set and an edge presents in the product whenever or (u 1 = u 2 and ). We investigate the sufficient conditions for vertex pancyclicity of lexicographic products of complete graphs Kn , paths Pn or cycles Cn with a general graph. We obtain that (i) if G 1 is a traceable graph of even order and G 2 is a graph with at least one edge, then is vertex pancyclic; (ii) if G 1 is hamiltonian and G 2 is a graph with at least one edge, then is vertex pancyclic.

For an undirected graph G¿=¿(V,E), we say that for l,u,v¿¿¿V, l separates u from v if the distance between u and l differs from the distance from v to l. A set of vertices L¿¿¿V is a feasible solution if for every pair of vertices u,v¿¿¿V there is l¿¿¿L that separates u from v. The metric dimension of a graph is the minimum cardinality of such a feasible solution. Here, we extend this well-studied problem to the case where each vertex v has a non-negative cost, and the goal is to find a feasible solution with a minimum total cost. This weighted version is NP-hard since the unweighted variant is known to be NP-hard. We show polynomial time algorithms for the cases where G is a path, a tree, a cycle, a cograph, a k-edge-augmented tree (that is, a tree with additional k edges) for a constant value of k, and a (not necessarily complete) wheel. The results for paths, trees, cycles, and complete wheels extend known polynomial time algorithms for the unweighted version, whereas the other results are the first known polynomial time algorithms for these classes of graphs even for the unweighted version. Next, we extend the set of graph classes for which computing the unweighted metric dimension of a graph is known to be NPC by showing that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the unweighted metric dimension of the graph is NPC.

The line graph L(G) of G has E(G) as its vertex set, and two vertices are adjacent in L(G) if and only if the corresponding edges share a common end vertex in G. Let σ¯2(G)=min{dG(x)+dG(y)|xy∈E(G)}. We show that, if σ¯2(G)≥2(⌊n11⌋−1) and n is sufficiently large, then either L(L(G)) is traceable or the Veldman’s reduction G′ is one of well-defined classes of exceptional graphs. Furthermore, if σ¯2(G)≥2(⌊n7⌋−1) and n is sufficiently large, then L(L(G)) is traceable. The bound 2(⌊n7⌋−1) is sharp. As a byproduct, we characterize the structure of a connected graph with a non-traceable 2-iterated line graph.

An Ore-type condition for the existence of two disjoint cycles

Cyclability and pancyclability in bipartite graphs

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ⩾ 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ⩾ 2δ2, then for any 3 ⩽ l1 ⩽ l2 ⩽ ⃛ ⩽ lk ⩽ n, 1 = k = [(δ ‐ 1)/2], such graph has K edge disjoint cycles with lengths l1, l2…lk, respectively. In particular, when l1 = l2 = ⃛ = lk = n and k = [(δ ‐ 1)/2], the graph contains [(δ ‐ 1)/2] edge disjoint hamiltonian cycles.

Spectra of quasi-strongly regular graphs

Given n vertices and m edges, m ≥ 1, and for every vertex is given a label, there are lots of graphs that can be obtained. The graphs obtained may be simple or not simple, connected or disconnected. A graph G(V,E) is called simple if G(V,E) not containing loops nor paralel edges. An edge which has the same end vertex is called a loop, and paralel edges are two or more edges which connect the same set of vertices. Let N(G7,m,t) as the number of connected vertex labeled graphs of order seven with m vertices and t (t is the number edges that connect different pair of vertices). The result shows that N(G7,m,t) = ct C (m−1) t−1, with c6=6727, c7=30160 , c8=30765, c9=21000, c10=28364, c11=26880, c12=26460, c13=20790, c14=10290, c15= 8022, c16=2940, c17=4417, c18=2835, c19=210, c20= 21, c21=1.

A new condition on dominated pair degree sum for a digraph to be supereulerian

On independent generalized degrees and independence numbers in K(1, m)-free graphs

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n 1, . . . , n k ) of positive integers such that n 1 + · · · + n k = n there exists a partition (V 1, . . . , V k ) of the vertex set of G such that for each \({i \in \{1,\ldots,k\}}\) , V i induces a connected subgraph of G on n i vertices. The main result of the paper reads as follows. Suppose that G is a connected graph on n ≥ 20 vertices that admits a perfect matching or a matching omitting exactly one vertex. If the degree sum of any pair of nonadjacent vertices is at least n − 5, then G is arbitrarily vertex decomposable. We also describe 2-connected arbitrarily vertex decomposable graphs that satisfy a similar degree sum condition.

A graph is denoted by $G$ with the vertex set $V(G)$ and the edge set $E(G)$.A path $P=\langle v_0,v_1,\cdots,v_m\rangle$ is a sequence of adjacent vertices. Two paths with equal length $P_1=\langle u_1,u_2,\ldots,u_m\rangle$ and $P_2=\langle v_1,v_2,\ldots,v_m\rangle$ from $a$ to $b$ are {\it independent} if $u_1=v_1=a$, $u_m=v_m=b$, and $u_i\neq v_i$ for $2\leq i\leq m-1$.Paths with equal length $\{P_i\}^{n}_{i=1}$ from $a$ to $b$ are {\it mutually independent} if they are pairwisely independent. Let $u$ and $v$ be two distinct vertices of a bipartite graph $G$, and let $l$ be a positive integer length, $d_G(u,v)\le l \le |V(G)-1|$ with $(l-d_G(u,v))$ being even. We say that the pair of vertices $u$, $v$ is {\it $(m,l)$-mutually independent bipanconnected} if there exist $m$ mutually independent paths $\{P_i^l\}_{i=1}^m$ with length $l$ from $u$ to $v$. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices $u$ and $v$ in the $n$-dimensional hypercube, with $d_{Q_n}(u,v) \ge n-1$, is $(n-1,l)$-mutually independent bipanconnected for every $l$, $d_{Q_n}(u,v) \le l \le |V(Q_n)-1|$ with $(l-d_{Q_n}(u,v))$ being even. As for $d_{Q_n}(u,v) \le n-2$, it is also $(n-1,l)$-mutually independent bipanconnected if $l \ge d_{Q_n}(u,v)+2$, and is only $(l,l)$-mutually independent bipanconnected if $l=d_{Q_n}(u,v)$.

Let G be a finite abelian group with identity 0. For an integer the additive power graph of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some with When the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by More specifically, we obtain the diameter and the girth of the graph and its complement Using these, we obtain a condition for and its complement to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of and its complement and hence we prove that and its complement are weakly perfect. Also, we discuss about the perfectness of At last, we obtain a condition for and its complement to be vertex pancyclic.

Degree sums and proper connection number of graphs

The complete twisted graph of order n, denoted by $$T_n$$ , is a complete simple topological graph with vertices $$u_1, u_2, \ldots , u_n$$ such that two edges $$u_iu_j$$ and $$u_{i'}u_{j'}$$ cross if and only if $$ i<i'<j'<j $$ or $$ i'<i<j<j' $$ . The convex geometric complete graph of order n, denoted by $$G_n$$ , is a convex geometric graph with vertices $$v_1, v_2, \ldots , v_n$$ placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set $$\{v_1,v_2,\ldots , v_n \}$$ having the property of being planar when embedded in both $$T_n$$ and $$G_n$$ . Given a connected graph G the (combinatorial) tree graph $$\mathcal {T}(G)$$ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in $$\mathcal {T}(G)$$ if there are edges $$e \in P$$ and $$f\in Q$$ such that $$Q=P - e + f$$ . For all positive integers n, we denote by $$\mathcal {T}(n)$$ the graph $$\mathcal {T}(K_n)$$ . The biplanar tree graph, $$\mathcal {B}(n)$$ , is the subgraph of $$\mathcal {T}(n)$$ induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.