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     

The number of contractible edges in 3-connected graphs

An edge of a 3-connected graph is calledcontractible if its contraction results in a 3-connected graph. Ando, Enomoto and Saito proved that every 3-connected graph of order at least five has |G|/2 or more contractible edges. As another lower bound, we prove that every 3-connected graph, except for six graphs, has at least (2|E(G)| + 12)/7 contractible edges. We also determine the extremal graphs. Almost all of these extremal graphsG have more than |G|/2 contractible edges.     

Covering contractible edges in 3-connected graphs. II. Characterizing those with covers of size three

It is shown that if G is a 3-connected graph with |V(G)| ≥ 10, then, with the exception of one infinite class based on K_3,p, it takes at least four vertices to cover the set of contractible edges of G. © 1993 John Wiley & Sons, Inc.     

Maximum Number of Contractible Edges on Hamiltonian Cycles of a 3-Connected Graph

 We show that if G is a 3-connected hamiltonian graph of order at least 5, then there exists a hamiltonian cycle C of G such that the number of contractible edges of G which are on C is greater than or equal to . Received: July 31, 2000 Final version received: December 12, 2000 Acknowledgments. I would like to thank Professor Yoshimi Egawa for the help he gave to me during the preparation of this paper.     

Removable Edges in Longest Cycles of 4-Connected Graphs

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G–e; second, for all vertices x of degree 3 in G–e, delete x from G–e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by Ge. If Ge is 4-connected, then e is called a removable edge of G. In this paper we obtain some results on removable edges in a longest cycle of a 4-connected graph G. We also show that for a 4-connected graph G of minimum degree at least 5 or girth at least 4, any edge of G is removable or contractible.Acknowledgment. The authors are greatly indebted to a referee for his valuable suggestions and comments, which are very helpful to improve the proof of our main result Lemma 3.3.Research supported by National Science Foundation of China AMS subject classification (2000): 05C40, 05C38, 05C75Final version received: March 10, 2004     

On the Structure of Contractible Edges in <Emphasis Type="Italic">k</Emphasis>-connected Partial <Emphasis Type="Italic">k</Emphasis>-trees

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k − 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees.     

The 3-connected graphs with a maximum matching containing precisely one contractible edge

It has previously been shown that if M is a maximum matching in a 3-connected graph G, other than K₄, then M contains at least one contractible edge of G. In this paper, we give a constructive characterization of the 3-connected graphs G having a maximum matching containing only one contractible edge of G.     

More on foxes

An edge in a k‐connected graph G is called k‐contractible if the graph obtained from G by contracting e is k‐connected. Generalizing earlier results on 3‐contractible edges in spanning trees of 3‐connected graphs, we prove that (except for the graphs if ) (a) every spanning tree of a k‐connected triangle free graph has two k‐contractible edges, (b) every spanning tree of a k‐connected graph of minimum degree at least has two k‐contractible edges, (c) for , every DFS tree of a k‐connected graph of minimum degree at least has two k‐contractible edges, (d) every spanning tree of a cubic 3‐connected graph nonisomorphic to K₄ has at least many 3‐contractible edges, and (e) every DFS tree of a 3‐connected graph nonisomorphic to K₄, the prism, or the prism plus a single edge has two 3‐contractible edges. We also discuss in which sense these theorems are best possible.     

Contractible subgraphs in k‐connected graphs

For a graph G we define a graph T(G) whose vertices are the triangles in G and two vertices of T(G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k‐connected graph and T(G) contains no edge, then G admits a k‐contractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashi's result by showing that if G is k‐connected and the maximum degree of T(G) is at most 1, then G admits a k‐contractible clique of size at most 3 or there exist independent edges e and f of G such that e and f are contained in triangles sharing an edge and G/e/f is k‐connected. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 121–136, 2007     

Every DFS Tree of a 3‐Connected Graph Contains a Contractible Edge

Tutte proved that every 3‐connected graph G on more than 4 vertices contains a contractible edge. We strengthen this result by showing that every depth‐first‐search tree of G contains a contractible edge. Moreover, we show that every spanning tree of G contains a contractible edge if G is 3‐regular or if G does not contain two disjoint pairs of adjacent degree‐3 vertices.     

Double cycle covers and the petersen graph

Let O(G) denote the set of odd-degree vertices of a graph G. Let t ? N and let ??_t denote the family of graphs G whose edge set has a partition. E(g) = E₁ U E₂ U … U E_tsuch that O(G) = O(G[E_i]) (1 ? i ? t). This partition is associated with a double cycle cover of G. We show that if a graph G is at most 5 edges short of being 4-edge-connected, then exactly one of these holds: G ? ??₃, G has at least one cut-edge, or G is contractible to the Petersen graph. We also improve a sufficient condition of Jaeger for G ? ??_2p+1(p ? N).     

3-connected graphs with non-cut contractible edge covers of size k

In this paper, we show that if a 3-connected graph G other than K₄ has a vertex subset K that covers the set of contractible edges of G and if |K| 3 and |V(G)| 3|K| ? 1, then K is a cutset of G. We also give examples to show that this result is best possible. In particular, the result does not hold for K with smaller cardinality.     

Cycle and cocycle coverings of graphs

In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family ?? of at most n?1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and ε edges having cogirth g^*?3 and k(G) components, there is a family of at most ε?n+k(G) cocycles which cover the edges of G at least twice. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 270–284, 2010     

Wing-triangulated graphs are perfect

The wing-graph W(G) of a graph G has all edges of G as its vertices; two edges of G are adjacent in W(G) if they are the nonincident edges (called wings) of an induced path on four vertices in G. Hoàng conjectured that if W(G) has no induced cycle of odd length at least five, then G is perfect. As a partial result towards Hoàng's conjecture we prove that if W(G) is triangulated, then G is perfect. © 1997 John Wiley & Sons, Inc.     

Long cycles in subgraphs of (pseudo)random directed graphs

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directed cycle of length at least (c ? o(1))n. Moreover, there is a subgraph G′′of G with (1/2 + γ ? o(1))|E| edges that does not contain a cycle of length at least cn. © 2011 Wiley Periodicals, Inc. J Graph Theory 70: 284–296, 2012     

Covering contractible edges in 3-connected graphs. I: Covers of size three are cutsets

An edge of a 3-connected graph is said to be contractible if its contraction results in a 3-connected graph. In this paper, a covering of contractible edges is studied. We give an alternative proof to the result of Ota and Saito (Scientia (A) 2 (1988) 101–105) that the set of contractible edges in a 3-connected graph cannot be covered by two vertices, and extended this result to a three-vertex covering. We also study the existence of a contractible edge whose contraction preserves a specified cycle, and show that a non-hamiltonian 3-connected graph has a contractible edge whose contraction preserves the circumference.     

Estimates of AGMON-MIRANDA type for solutions of the DIRICHLET problem in plane domains with corners

It is proved that any edge of a 4-connected non-planar graph G of order at least 6 lies in a subdivision of K_3,3 in G. For any 3-connected non-planar graph G of order at least 6 we show that G contains at most four edges which belong to no subdivisions of K_3,3 in G.     

Contractions to k8

It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K₈ is 6n ? 21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n ? 21 edges that are not contractible to K₈ are the K₅(2)-cockades that have exactly 6n ? 20 edges.     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Acyclic total colorings of planar graphs without <Emphasis Type="Italic">l</Emphasis> cycles

A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without l cycles is at most Δ(G) + 2 if (k, l) ∈ {(6, 3), (7, 4), (6, 5), (7, 6)}.