Extension of a graph G is to construct a new graph with certain properties by adding edges to some pairs of vertices in G. In this paper, we focus on extension of a quadrangulation of a surface to even triangulations, where a quadrangulation is a map on a surface with every face quadrangular and a triangulation is even if all the vertices have even degree. Zhang and He [SIAM J. Comput. 34 (2005...

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two di erent degrees.

Let H be a hypergraph with n vertices. Suppose that d1, d2, . . . , dn are degrees of the vertices of H. The t-th graph entropy based on degrees of H is defined as

We present theoretical and computational results onα-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of α-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise...

We study the problem of constructing a (near) random proper q-colouring of a simple k-uniform hypergraph with n vertices and maximum degree ∆. (Proper in that no edge is mono-coloured and simple in that two edges have maximum intersection of size one). We give conditions on q,∆ so that if these conditions are satisfied, Glauber dynamics will converge in O(n log n) time from a random (improper) ...

Here we denote a diameter six tree by (c; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where c is the center of the tree; ai, i = 1, 2, . . . ,m, bj, j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to c; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful lab...

Let G be a graph with vertex set V (G). Let n; k and d be non-negative integers such that n+2k+d6 |V (G)|−2 and |V (G)|−n−d is even. A matching which covers exactly |V (G)|−d vertices of G is called a defect-d matching of G. If when deleting any n vertices of G the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect-d matching, then G is called a (...

We investigate issues surrounding an old question of Yates’ as to the existence of a minimal degree with no strong minimal cover, specifically with respect to the hyperimmune-free degrees.

For a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f2(G) ≥ √ n(1− o(1)), where n is the number of vertices of G. It is conjectured that fk(G) = Θ( √ n) for every fixed k. We prove this for k = 2,...

For a given N-vertex graph. H, a graph G obtained from H by adding t vertices a,nd some edges is called a t-FT (t-fault-toleran,t) graph. for H if even after deletin.g any t vertices from G, the remaining graph contains H as a subgraph. For an N-vertex hypercube QN, a t-FT graph with an, optimal number O(tN +t2) of added edges and maximum degree of O(N + t), and a, t-FT graph zuith O(tN log N) ...