We prove that every oriented tree of order n ≥ 5 with three leaves is (n + 1)-unavoidable. More precisely, we prove that every tree A of order n with three leaves is contained in every tournament T of order n + 1 except if T is the regular tournament on five vertices and A the...
The studies of such distance-based graph invariants include, for instance, the "middle part" of a tree, the extremal structures with given constraints, the extremal values of ratios of the distance function at the "middle part" and leaves. In particular, when considering the extremal structures...
1 contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw's result for connected K1,5-free graphs. We show that every n-vertex connected K1,5-free graph G with σ5(G)≥n?1 contains a spanning tree with at most 4 leaves. Moreover, the degree...
Tree is a connected graph without cycles. A leaf of a tree is any vertex connected with exactly one other vertex. You are given a tree with n vertices and a root in the vertex 1. There is an ant in each leaf of the tree. In one second some ants can simultaneously go to the paren...
Every connected K1,4-free graph G with σ4(G)≥|G|−1 contains a spanning tree with at most 3 leaves. 2. Proof of Theorem A To prove Theorem A, we need the following lemma. Lemma 1 Suppose that G does not have a spanning tree with at most 3 leaves. Let T be a maximal tree...
Summary: We present two lower bounds for the maximum number of leaves over all spanning trees of a graph. For connected, triangle-free graphs on $n$ vertices, with minimum degree at least three, we show that a spanning tree with at least $(n+4)/3$ leaves exists. For connected graphs...
great grand child of root grand children of root children of root Example Tree President VP1VP2 VP3 Manager1Manager2Manager Worker Bee root Definition A tree t is a finite nonempty set of elements. One of these elements is called the root. The remaining elements, if any, are partitioned ...
It was shown by Griggs and Wu that a graph of minimal degree 4 on n vertices has a spanning tree with at least \frac25 n leaves, which is asymptomatically the best possible bound for this class of graphs. In this paper, we present a polynomial time algorithm that finds in any graph ...
We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for all graphs besides three ...
Tree is a connected graph without cycles. A leaf of a tree is any vertex connected with exactly one other vertex. You are given a tree withnvertices and a root in the vertex1. There is an ant in each leaf of the tree. In one second some ants can simultaneously go to the parent ve...