A spanning subgraph F of a graph G is called perfect if F is a forest, the degree dF (x) of each vertex x in F is odd, and each tree of F is an induced subgraph of G. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph G contains a perfect forest if and only if G has an even number of vertices.

Let H be a multigraph and G a graph containing a subgraph isomorphic to a subdivision of H, with S ⊂ V (G) (the ground set) the image of V (H) under the isomorphism. We consider connectivity and minimum degree or degree sum conditions sufficient to imply there is a spanning subgraph of G isomorphic to a subdivision of H on the same ground set. These results generalize a number of theorems in th...

For a fixed integer m, we consider edge colorings of complete graphs which contain no properly edge colored cycle Cm as a subgraph. Within colorings free of these subgraphs, we establish a global structure by bounding the number of colors that can induce a spanning and connected subgraph. In the case of small cycles, namely C4, C5, and C6, we show that our bounds are sharp.

Let G be a graph and f : V (G) → {2, 3, . . .}. A spanning subgraph F is called strong f -star of G if each component of F is a star whose center x satisfies degF (x) ≤ f(x) and F is an induced subgraph of G. In this paper, we prove that G has a strong f -star factor if and only if oddca(G − S) ≤

Let G =< V,E > be a connected graph with real-valued edge weights: w : E → R, having n vertices and m edges. A spanning tree in G is an acyclic subgraph of G that includes every vertex of G and is connected; every spanning tree has exactly n− 1 edges. A minimum spanning tree (MST) is a spanning tree of minimum weight which is defined to be the sum of the weights of all its edges. Our problem is...

We prove that every (6k + 2`, 2k)-connected simple graph contains k rigid and ` connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach ...

A k-tree is either a complete graph on k vertices or a graph T that contains a vertex whose neighbourhood in T induces a complete graph on k vertices and whose removal results in a k-tree. A subgraph of a graph is a spanning k-tree if it is a k-tree and contains every vertex of the graph. This paper is concerned with spanning 2-trees in a graph. It is shown that spanning 2-trees have close conn...

Proof to Lemma 2. First, yi ≥ 0. For each i, we construct |V | affinely independent points in C with yi = 0. Fix i, then one solution is obviously x = 0, the empty subgraph. Next, for all p 6= i, obtain one solution by setting only yp = 1, and for all j 6= p set yj = 0. Clearly, yj = 0 and the |V | − 1 solutions thus obtained are affinely independent. In total we have |V | solutions with yi = 0...

The MAXIMUM PLANAR SUBGRAPH problem|given a graph G, nd a largest planar subgraph of G|has applications in circuit layout, facility layout, and graph drawing. No previous polynomial-time approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1=3, which is achieved simply by producing a spanning tree of G. We present the rst approximation algori...

We prove that every (6k + 2`, 2k)-connected simple graph contains k rigid and ` connected edgedisjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach p...