Independent trees in graphs

IfG is a finite undirected graph ands is a vertex ofG, then two spanning treesT ₁ andT ₂ inG are calleds — independent if for each vertexx inG the paths fromx tos inT ₁ andT ₂ are openly disjoint. It is known that the following statement is true fork3: IfG isk-connected, then there arek pairwises — independent spanning, trees inG. As a main result we show that this statement is also true fork=4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weaklys — independent spanning trees (i.e., the tree paths from a vertex tos are edge disjoint) and for directed graphs.     

Independent Trees in Planar Graphs Independent trees

We prove the following statement: Let G be a finite k-connected undirected planar graph and s be a vertex of G. Then there exist k spanning trees T₁,…,T_k in G such that for each vertex xps of G, the k paths from x to s in T₁,…,T_k are pairwise openly disjoint.     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs

Given a vertex r of a 3-connected graph G, we show how to find three independent spanning trees of G rooted at r. Our proof is based on showing that every 3-connected graph has a nonseparating ear decomposition. This extends Whitney's characterisation that a graph is 2-connected iff it has an ear decomposition. We also show that a nonseparating ear decomposition can be constructed in O(VE) time, and hence, three independent spanning trees can be found in O(VE) time. We construct a nonseparating ear decomposition by solving the following problem at most V times. Given an edge tr and a vertex u of a 3-connected graph G, find a nonseparating induced cycle of G through tr and avoiding u. W. T. Tutte (Proc. London Math. Soc. 13 (1963), 743–767) first showed that such a cycle can always be found. We give a linear time algorithm for this.     

Removable edges in cycles of a <Emphasis Type="Italic">k</Emphasis>-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N _G−e (x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4).     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

Parity vertex colouring of plane graphs

A proper vertex colouring of a 2-connected plane graph G is a parity vertex colouring if for each face f and each colour c, either no vertex or an odd number of vertices incident with f is coloured with c. The minimum number of colours used in such a colouring of G is denoted by χ_p(G).In this paper, we prove that χ_p(G)≤118 for every 2-connected plane graph G.     

A note on locally connected and hamiltonian-connected graphs

A graphG is locallyn-connected,n≧1, if the subgraph induced by the neighborhood of each vertex isn-connected. It is shown that every connected, locally 3-connected graph containing no induced subgraph isomorphic toK(1, 3) is hamiltonian-connected.     

Independent Trees and Branchings in Planar Multigraphs

 The following statement is proved: Let G be a finite directed or undirected planar multigraph and s be a vertex of G such that for each vertex x≠s of G, there are at least k pairwise openly disjoint paths in G from x to s where k∉{3,4,5} if G is directed. Then there exist k spanning trees T ₁, … ,T _k in G directed towards s if G is directed such that for each vertex x≠s of G, the k paths from x to s in T ₁, … ,T _k are pairwise openly disjoint. – The case where G is directed and k∈{3,4,5} remains open. Received: January 30, 1995 / Revised: October 7, 1996     

Connectivity of intersection graphs of finite groups

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H∩K≠1 where 1 denotes the trivial subgroup of G. In this paper, we classify finite solvable groups whose intersection graphs are not 2-connected and finite nilpotent groups whose intersection graphs are not 3-connected. Our methods are elementary.     

A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X|≥3 and if X is 4-connected in G. If X=V(G) then Sanders’ result follows.     

Parity vertex coloring of outerplane graphs

A proper vertex coloring of a 2-connected plane graph G is a parity vertex coloring if for each face f and each color c, the total number of vertices of color c incident with f is odd or zero. The minimum number of colors used in such a coloring of G is denoted by χ_p(G).In this paper we prove that χ_p(G)≤12 for every 2-connected outerplane graph G. We show that there is a 2-connected outerplane graph G such that χ_p(G)=10. If a 2-connected outerplane graph G is bipartite, then χ_p(G)≤8, moreover, this bound is best possible.     

Characterization of maximum critically 2-connected graphs

A graph G is critically 2-connected if G is 2-connected but, for any point p of G, G — p is not 2-connected. Critically 2-connected graphs on n points that have the maximum number of lines are characterized and shown to be unique for n ? 3, n ≠ 11.     

The Erdős–Pósa Property for Odd Cycles in Graphs of Large Connectivity

Dedicated to the memory of Paul Erdős A graph G is k-linked if G has at least 2k vertices, and, for any vertices , , ..., , , , ..., , G contains k pairwise disjoint paths such that joins for i = 1, 2, ..., k. We say that G is k-parity-linked if G is k-linked and, in addition, the paths can be chosen such that the parities of their lengths are prescribed. We prove the existence of a function g(k) such that every g(k)-connected graph is k-parity-linked if the deletion of any set of less than 4k-3 vertices leaves a nonbipartite graph. As a consequence, we obtain a result of Erdős–Pósa type for odd cycles in graphs of large connectivity. Also, every -connected graph contains a totally odd -subdivision, that is, a subdivision of in which each edge of corresponds to an odd path, if and only if the deletion of any vertex leaves a nonbipartite graph. Received May 13, 1999/Revised June 19, 2000     

An Extremal Problem on Contractible Edges in 3-Connected Graphs

An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu [7] proved that G has at most vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation. Research partially supported by an ONR grant under grant number N00014-01-1-0917     

Connected,locally 2-connected,K1,3-free graphs are panconnected

A graph G is locally n-connected, n ≥ 1, if the subgraph induced by the neighborhood of each vertex is n-connected. We prove that every connected, locally 2-connected graph containing no induced subgraph isomorphic to K_1,3 is panconnected.     

The Cendian of a Tree

The eccentricity e(x) and the distance sum s(x) of a vertex x of a connected graph G are well-known functions which measure the centrality of the vertex x in G. The set of vertices which minimize e(x) is called the center of G and the set of vertices which minimize s(x) is known as the median. In this paper we introduce the idea of the so-called cendian of a graph, which unifies the concepts of center and median, and study its structure in trees.     

On the computational complexity of ordered subgraph recognition

Let (G, <) be a finite graph G with a linearly ordered vertex set V. We consider the decision problem (G, <)ORD to have as an instance an (unordered) graph Γ and as a question whether there exists a linear order ≤ on V(Γ) and an order preserving graph isomorphism of (G, <) onto an induced subgraph of Γ. Several familiar classes of graph are characterized as the yes-instances of (G, < )ORD for appropriate choices of (G, <). Here the complexity of (G, <)ORD is investigated. We conjecture that for any 2-connected graph G, G ≠ K_k, (G, <)ORD is NP-complete. This is verified for almost all 2-connected graphs. Several related problems are formulated and discussed. © 1995 John Wiley & Sons, Inc.     

On the connectivity of maximal planar graphs

Let d₁ ? d₂ ? … ? d_p be the vertex degrees of a maximal planar graph G. Etourneau has shown that if d₁ ? 6 and d_p = 5, then G is 5-connected. We generalize Etourneau's result by giving sufficient conditions in terms of the vertex degrees for G to be d_p -connected.     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.