We prove that every YES instance of BALANCED ST-CONNECTIVITY [Kin10] has a balanced path of polynomial length.

Given a graph G and a positive integer r, let sr(G)=P(G; r)=r!. Thus (G)=r and sr(G)=1 i G is uniquely r-colourable. It is known that if G is uniquely 3-colourable, then e(G)¿2v(G)−3. In this paper, we show that if G is a 3-colourable connected graph with e(G)=2v(G)−k where k¿4, then s3(G)¿2k−3; and if, further, G is 2-connected and s3(G)=2k−3, then t(G)6v(G)−k where t(G) denotes the number of ...

We characterize those Baire one functions f for which the diagonal product x 7→ (f(x), g(x)) has a connected graph whenever g is approximately continuous or is a derivative.

Let T be any tree of order d ≥ 1. We prove that every connected graph G with minimum degree d contains a subtree T ′ isomorphic to T such that G − V (T ) is connected.

For a graph G, its rth power G is defined as the graph with the same vertex set as G, and an edge between any two vertices whenever they are within distance r of each other in G. Motivated by a result from additive number theory, Hegarty raised the question of how many new edges G has when G is a regular, connected graph with diameter at least r. We address this question for r = 3, 6. We give a...

The present note is devoted to establish some extremal results for the zerothorder general Randić index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.

The center of a connected graph G is the set of nodes of G for which the maximum distance to any other node of G is as small as possible. If G is a simply connected set of lattice points (\pixels") with graph structure deened by 4-neighbor adjacency, we show that the center of G is either a 2-by-2 square block, a diagonal staircase, or a (dotted) diagonal line with no gaps. The authors thank Fr...

An essentially k-edge connected graph G is a connected graph such that deleting less than k edges from G cannot result in two nontrivial components. In this paper we prove that if an essentially 2-edge-connected graph G satisfies that for any pair of leaves at distance 4 in G there exists another leaf of G that has distance 2 to one of them, then the square G2 has a connected even factor with m...

If G is a connected graph with vertex set V , then the degree distance of G, D′(G), is defined as ∑ {u,v}⊆V (deg u + deg v) d(u, v), where degw is the degree of vertex w, and d(u, v) denotes the distance between u and v. We prove the asymptotically sharp upper bound D′(G) ≤ 14 nd(n− d) 2 +O(n7/2) for graphs of order n and diameter d. As a corollary we obtain the bound D′(G) ≤ 1 27 n 4 + O(n7/2)...

We prove that a 4-connected K 4;4-minor free graph on n vertices has at most 4n ? 8 edges and we use this result to show that every K 4;4-minor free graph has vertex-arboricity at most 4. This improves the case (n; m) = (7; 3) of the following conjecture of Woodall: the vertexset of a graph without a K n-minor and without a K b n+1 2 c;d n+1 2 e-minor can be partitioned in n ? m + 1 subgraphs w...