6 research outputs found
Note on Perfect Forests
A spanning subgraph of a graph is called perfect if is a forest,
the degree of each vertex in is odd, and each tree of is
an induced subgraph of . We provide a short proof of the following theorem
of A.D. Scott (Graphs & Combin., 2001): a connected graph contains a
perfect forest if and only if has an even number of vertices
Note on Perfect Forests in Digraphs
A spanning subgraph of a graph is called {\em perfect} if is a
forest, the degree of each vertex in is odd, and each tree of
is an induced subgraph of . Alex Scott (Graphs \& Combin., 2001) proved
that every connected graph contains a perfect forest if and only if has
an even number of vertices. We consider four generalizations to directed graphs
of the concept of a perfect forest. While the problem of existence of the most
straightforward one is NP-hard, for the three others this problem is
polynomial-time solvable. Moreover, every digraph with only one strong
component contains a directed forest of each of these three generalization
types. One of our results extends Scott's theorem to digraphs in a non-trivial
way
Two short proofs of the Perfect Forest Theorem
A perfect forest is a spanning forest of a connected graph , all of whose components are induced subgraphs of and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra.
We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynomial-time algorithms for finding a perfect forest in a connected graph of even order
Perfect Forests in Graphs and Their Extensions
Let G be a graph on n vertices. For i ? {0,1} and a connected graph G, a spanning forest F of G is called an i-perfect forest if every tree in F is an induced subgraph of G and exactly i vertices of F have even degree (including zero). An i-perfect forest of G is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 0-perfect forest not containing e in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest
Maximum odd induced subgraph of a graph concerning its chromatic number
Let be the maximum order of an odd induced subgraph of . In
1992, Scott proposed a conjecture that for
a graph of order without isolated vertices, where is the
chromatic number of . In this paper, we show that the conjecture is not true
for bipartite graphs, but is true for all line graphs. In addition, we also
disprove a conjecture of Berman, Wang and Wargo in 1997, which states that
for a connected graph of order
. Scott's conjecture is open for a graph with chromatic number at least 3.Comment: 10 pages, 1 figur