Maximum chromatic polynomials of 2-connected graphs

In this paper we obtain chromatic polynomials P(G; λ) of 2-connected graphs of order n that are maximum for positive integer-valued arguments λ ≧ 3. The extremal graphs are cycles C_n and these graphs are unique for every λ ≧ 3 and n ≠ 5. We also determine max{P(G; λ): G is 2-connected of order n and G ≠ C_n} and all extremal graphs relative to this property, with some consequences on the maximum number of 3-colorings in the class of 2-connected graphs of order n having _X(G) = 2 and _X(G) = 3, respectively. For every n ≧ 5 and λ ≧ 4, the first three maximum chromatic polynomials of 2-connected graphs are determined.     

Matching extension in the powers of n-connected graphs

Let G be a graph on p vertices. Then for a positive integer n₁ G is said to be n-extendible if (i) n < p/2, (ii) G has a set of n independent edges, and (iii) every such set is contained in a perfect matching of G. In this paper we will show that if p is even and G is n-connected, then G^k is -extendible for every integer k ≥ 2 such that . © 1996 John Wiley & Sons, Inc.     

A new recursive theorem onn-extendibility

A graphG having a 1-factor is calledn-extendible if every matching of sizen extends to a 1-factor. LetG be a 2-connected graph of order 2p. Letr0 andn>0 be integers such thatp–rn+1. It is shown that ifG/S isn-extendible for every connected subgraphS of order 2r for whichG/S is connected, thenG isn-extendible.     

Generating planar 4-connected graphs

In this paper, we introduce three operations on planar graphs that we call face splitting, double face splitting, and subdivision of hexagons. We show that the duals of the planar 4-connected graphs can be generated from the graph of the cube by these three operations. That is, given any graphG that is the dual of a planar 4-connected graph, there is a sequence of duals of planar 4-connected graphsG ₀,G ₁, …,G _n such thatG ₀ is the graph of the cube,G _n=G, and each graph is obtained from its predecessor by one of our three operations. Research supported by a Sloan Foundation fellowship and by NSF Grant#GP-27963.     

Pancyclic and panconnected line graphs

Let G be a graph of order n ? 3. We prove that if G is k-connected (k ? 2) and the degree sum of k + 1 mutually independent vertices of G is greater than 1/3(k + 1)(n + 1), then the line graph L(G) of G is pancyclic. We also prove that if G is such that the degree sum of every 2 adjacent vertices is at least n, then L(G) is panconnected with some exceptions.     

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.     

Neighborhoods and Covering Vertices by Cycles

G =(V,E) is a 2-connected graph, and X is a set of vertices of G such that for every pair x,x' in X, , and the minimum degree of the induced graph <X> is at least 3, then X is covered by one cycle. This result will be in fact generalised by considering tuples instead of pairs of vertices. Let be the minimum degree in the induced graph <X>. For any , . If , and , then X is covered by at most (p-1) cycles of G. If furthermore , (p-1) cycles are sufficient. So we deduce the following: Let p and t () be two integers. Let G be a 2-connected graph of order n, of minimum degree at least t. If , and , then V is covered by at most cycles, where k is the connectivity of G. If furthermore , (p-1) cycles are sufficient. In particular, if and , then G is hamiltonian. Received April 3, 1998     

On Connected Transversals to Dihedral Subgroups of Order 2p n

Let G be a group with a dihedral subgroup H of order 2pⁿ, where p is an odd prime. We show that if there exist H-connected transversals in G, then G is a solvable group. We apply this result to the loop theory and show that if the inner mapping group of a finite loop Q is dihedral of order 2pⁿ, then Q is a solvable loop.1991 Mathematics Subject Classification: 20D10, 20N05     

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.     

On minors of graphs with at least 3n −4 edges

A point disconnecting set S of a graph G is a nontrivial m-separator, where m = |S|, if the connected components of G - S can be partitioned into two subgraphs, each of which has at least two points. A 3-connected graph is quasi 4-connected if it has no nontrivial 3-separators. Suppose G is a graph having n ≥ 6 points. We prove three results: (1) If G is quasi 4-connected with at least 3n - 4 edges, then the graph K^?₁, obtained from K₆ by deleting an edge, is a minor of G. (2) If G has at least 3n - 4 edges then either K^?₆ or the graph obtained by pasting two disjoint copies of K₅ together along a triangle is a minor of G. (3) If the minimum degree of G is at least 6, then K^?₆ is a minor of G. © 1993 John Wiley & Sons, Inc.     

One sufficient condition for Hamiltonian graphs involving distances

In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x) + d(y) ≥ 2n − 1 for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x)+d(y) ≥ 2n−1 for each pair of nonadjacent vertices x, y ∈ V (G) such that d(x, y) = 2, then G is Hamiltonian.     

High Connectivity Keeping Sets In <Emphasis Type="Italic">n</Emphasis>-Connected Graphs

It is proved that for every positive integer k, every n-connected graph G of sufficiently large order contains a set W of k vertices such that G—W is (n-2)-connected. It is shown that this does not remain true if we add the condition that G(W) is connected.     

On Whitney's 2-isomorphism theorem for graphs

Let G and H be 2-connected 2-isomorphic graphs with n nodes. Whitney's 2-isomorphism theorem states that G may be transformed to a graph G* isomorphic to H by repeated application of a simple operation, which we will term “switching”. We present a proof of Whitney's theorem that is much shorter than the original one, using a graph decomposition by Tutte. The proof also establishes a surprisingly small upper bound, namely n-2, on the minimal number of switchings required to derive G* from G. The bound is sharp in the sense that for any integer N there exist graphs G and H with n ≥ N nodes for which the minimal number of switchings is n-2.     

Toughness and longest cycles in 2-connected planar graphs

Let G be a planar graph on n vertices, let c(G) denote the length of a longest cycle of G, and let w(G) denote the number of components of G. By a well-known theorem of Tutte, c(G) = n (i.e., G is hamiltonian) if G is 4-connected. Recently, Jackson and Wormald showed that c(G) ≥ βn^α for some positive constants β and α ≅ 0.2 if G is 3-connected. Now let G have connectivity 2. Then c(G) may be as small as 4, as with K_2,n-2, unless we bound w(G − S) for every subset S of V(G) with |S| = 2. Define ξ(G) as the maximum of w(G − S) taken over all 2-element subsets S ⊆ V(G). We give an asymptotically sharp lower bound for the toughness of G in terms of ξ(G), and we show that c(G) ≥ θ ln n for some positive constant θ depending only on ξ(G). In the proof we use a recent result of Gao and Yu improving Jackson and Wormald's result. Examples show that the lower bound on c(G) is essentially best-possible. © 1996 John Wiley & Sons, Inc.     

A class of Hamiltonian regular graphs

In this paper, we show that n ? 4 and if G is a 2-connected graph with 2n or 2n?1 vertices which is regular of degree n?2, then G is Hamiltonian if and only if G is not the Petersen graph.     

Decomposition of 3-connected graphs

Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K _3,n for somen3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.     

Connectivity keeping edges in graphs with large minimum degree

The old well-known result of Chartrand, Kaugars and Lick says that every k-connected graph G with minimum degree at least 3k/2 has a vertex v such that G−v is still k-connected. In this paper, we consider a generalization of the above result [G. Chartrand, A. Kaigars, D.R. Lick, Critically n-connected graphs, Proc. Amer. Math. Soc. 32 (1972) 63–68]. We prove the following result:Suppose G is a k-connected graph with minimum degree at least 3k/2+2. Then G has an edge e such that G−V(e) is still k-connected.The bound on the minimum degree is essentially best possible.     

Cycles through a prescribed vertex set in N-connected graphs

A well-known result of Dirac (Math. Nachr. 22 (1960) 61) says that given n vertices in an n-connected G, G has a cycle through all of them. In this paper, we generalize Dirac's result as follows:Given at most vertices in an n-connected graph G when n3 and , then G has a cycle through exactly n vertices of them.This improves the previous known bound given by Kaneko and Saito (J. Graph Theory 15(6) (1991) 655).     

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.     

n-connectedness in pure 2-complexes

A graph is a 1-dimensional simplicial complex. In this work we study an interpretation of “n-connectedness” for 2-dimensional simplicial complexes. We prove a 2-dimensional analogue of a theorem by Whitney for graphs: Theorem (A Whitney type theorem for pure 2-complexes).Let G be a pure 2-complex with no end-triangles. Then G is n-connected if and only if the valence of e is at least n for every interior edge e of G, and there does not exist a juncture set J of less than n edges of G. Examples ofn-connected pure 2-complexes are then given, and some consequences are proved.