On the number of hamiltonian cycles in a maximal planar graph

We consider the problem of the minimum number of Hamiltonian cycles that could be present in a Hamiltonian maximal planar graph on p vertices. In particular, we construct a p-vertex maximal planar graph containing exactly four Hamiltonian cycles for every p ≥ 12. We also prove that every 4-connected maximal planar graph on p vertices contains at least p/(log₂ p) Hamiltonian cycles.     

Large non-planar graphs and an application to crossing-critical graphs

We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K4,k, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices.     

On the toughness index of planar graphs

The toughness indexτ(G) of a graph G is defined to be the largest integer t such that for any S ? V(G) with |S| > t, c(G - S) < |S| - t, where c(G - S) denotes the number of components of G - S. In particular, 1-tough graphs are exactly those graphs for which τ(G) ≥ 0. In this paper, it is shown that if G is a planar graph, then τ(G) ≥ 2 if and only if G is 4-connected. This result suggests that there may be a polynomial-time algorithm for determining whether a planar graph is 1-tough, even though the problem for general graphs is NP-hard. The result can be restated as follows: a planar graph is 4-connected if and only if it remains 1-tough whenever two vertices are removed. Hence it establishes a weakened version of a conjecture, due to M. D. Plummer, that removing 2 vertices from a 4-connected planar graph yields a Hamiltonian graph.     

Coloring 4‐cycle systems

An m‐cycle system of order n is a partition of the edges of the complete graph K_n into m‐cycles. We investigate k‐colorings of 4‐cycle systems in which no 4‐cycle is monochromatic. For any k ≥ 3, we construct a k‐chromatic 4‐cycle system. We also show that for any k ge; 2, there exists an integer w_k such that for all admissible n ≥ w_k, there is a k‐chromatic 4‐cycle system of order n. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Subgraphs with Restricted Degrees of Their Vertices in Planar 3-Connected Graphs

We have proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path on k vertices; the bound 5k is the best possible. Moreover, for every connected planar graph H other than a path and for every integer m ≥ 3 there is a 3-connected planar graph G such that each copy of H in G contains a vertex of degree at least m.     

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     

Vertices of Degree 5 in a Contraction Critically 5-connected Graph

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is said to be contraction critically k-connected. We prove that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5. We also show that, for a graph G and an integer k greater than 4, there exists a contraction critically k-connected graph which has G as its induced subgraph.     

Paths with restricted degrees of their vertices in planar graphs

In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23. Analogous results are stated for 3-connected planar graphs of minimum degree 4 and 5. Moreover, for every pair of integers n 3, k 4 there is a 2-connected planar graph such that every path on n vertices in it has a vertex of degree k.     

Connectivity of cages

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer v such there exists a (k; g)-graph with v vertices. A (k; g)-cage is a (k; g)-graph with f(k; g) vertices. In this paper we prove that the cages are monotonic in that f(k; g₁) < f(k; g₂) for all k ≥ 3 and 3 ≥ g₁ < g₂. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k. © 1997 John Wiley & Sons, Inc.     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

On the number of cycles of length 4 in a maximal planar graph

Let p and C₄ (G) be the number of vertices and the number of 4-cycles of a maximal planar graph G, respectively. Hakimi and Schmeichel characterized those graphs G for which C₄ (G) = 1/2(p² + 3p - 22). This characterization is correct if p ≥ 9. However, for p = 7 or 8, there is exactly one other graph which violates the theorem in the sense that the upper bound of C₄ (G) is also attained.     

Cycles intersecting a prescribed vertex set

A graph is said to have property P(k,l)(k ? l) if for any X ∈ (^G_k) there exists a cycle such that |X ∩ V(C)| = l. Obviously an n-connected graph (n ? 2) satisfies P(n,n). In this paper, we study parameters k and l such that every n-connected graph satisfies P(k,l). We show that for r = 1 or 2 every n-connected graph satisfies P(n + r,n). For r = 3, there are infinitely many 3-connected graphs that do not satisfy P(6,3). However, if n ? max{3,(2r ?1)(r + 1)}, then every n-connected graph satisfies P(n + r,n).     

A Sharp Threshold for k‐Colorability

Let k be a fixed integer and f_k(n, p) denote the probability that the random graph G(n, p) is k‐colorable. We show that for k≥3, there exists d_k(n) such that for any ϵ>0, (1) As a result we conclude that for sufficiently large n the chromatic number of G(n, d/n) is concentrated in one value for all but a small fraction of d>1. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 63–70, 1999     

Contractible Edges in k-Connected Graphs with Some Forbidden Subgraphs

In 2001, Kawarabayashi proved that for any odd integer k ≥ 3, if a k-connected graph G is \({K^{-}_{4}}\) -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ≥ 4, if a k-connected graph G is \({K_{4}^{-}}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ≥ 3, k ≥ 2t – 1 be integers. If a k-connected graph G is \({(K_{1}+(K_{2} \cup K_{1, t}))}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.     

Ramsey theory for graph connectivity

r_c(k) Denotes the smallest integer such that any c-edge-coloring of the r_c(k) vertex complete graph has a monochromatic k-connected subgraph. For any c, k ≧ 2, we show 2c(k – 1) + 1 ≦ r_c(k) < 10/3 c(k – 1) + 1, and further that 4(k – 1) + 1 ≧ r₂(k) < (3 + √ (k – 1) + 1. Some exact values for various Ramsey connectivity numbers are also computed.     

Cubic graphs with three Hamiltonian cycles are not always uniquely edge colorable

The generalized Petersen graph P(6k + 3, 2) has exactly 3 Hamiltonian cycles for k ≥ 0, but for k ≥ 2 is not uniquely edge colorable. This disproves a conjecture of Greenwell and Kronk [1].     

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.     

Defective coloring revisited

A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)- and (5, 1)-colorable, and that a graph of genus γ is (χ_γ/(d + 1) + 4, d)-colorable, where χ_γ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2, k)-coloring, for k ≥ 1, and the (3, 1)-coloring problems are NP-complete even for planar graphs. In general graphs (k, d)-coloring is NP-complete for k ≥ 3, d ≥ 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented. © 1997 John Wiley & Sons, Inc.     

On planar hypohamiltonian graphs

We present a planar hypohamiltonian graph on 42 vertices and (as a corollary) a planar hypotraceable graph on 162 vertices, improving the bounds of Zamfirescu and Zamfirescu and show some other consequences. We also settle the open problem whether there exists a positive integer N, such that for every integer n≥N there exists a planar hypohamiltonian/hypotraceable graph on n vertices. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 55‐68, 2011