Old and new generalizations of line graphs.
On -factors in the cube of a graph
On -connected, planar -almost pancyclic graphs
On a bound on algebraic connectivity: the case of equality
In a recent paper the authors proposed a lower bound on , where , , is an eigenvalue of a transition matrix of an ergodic Markov chain. The bound, which involved the group inverse of , was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when...
On a conjecture concerning the Petersen graph.
On connected resolving decompositions in graphs
For an ordered -decomposition of a connected graph and an edge of , the -code of is the -tuple , where is the distance from to . A decomposition is resolving if every two distinct edges of have distinct -codes. The minimum for which has a resolving -decomposition is its decomposition dimension . A resolving decomposition of is connected if each is connected for . The minimum for which has a connected resolving -decomposition is its connected decomposition...
On dissections of the n-cube
On distance local connectivity and vertex distance colouring
In this paper, we give some sufficient conditions for distance local connectivity of a graph, and a degree condition for local connectivity of a k-connected graph with large diameter. We study some relationships between t-distance chromatic number and distance local connectivity of a graph and give an upper bound on the t-distance chromatic number of a k-connected graph with diameter d.
On domination in 2-connected cubic graphs.
On equality of edge-connectivity and minimum degree of a graph
On k-closure operators in graphs
On k-closure operators in graphs. II
On k-Path Pancyclic Graphs
For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G...
On L-ideal-based L-zero-divisor graphs
In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.
On locally quasiconnected graphs and their upper embeddability
On Longest Cycles in Essentially 4-Connected Planar Graphs
A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that [...] . For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least ¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover,...
On minimum locally -(arc)-strong digraphs
On pancyclic line graphs
On rainbow connection.
On Separating Sets of Edges in Contraction-Critical Graphs.