New constructions of hypohamiltonian and hypotraceable graphs

A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable, but all vertex‐deleted subgraphs of G are hamiltonian/traceable. All known hypotraceable graphs are constructed using hypohamiltonian graphs; here we present a construction that uses so‐called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex‐deleted subgraphs are hamiltonian with exactly one exception, see [15]). This construction is an extension of a method of Thomassen [11]. As an application, we construct a planar hypotraceable graph of order 138, improving the best‐known bound of 154 [8]. We also prove a structural type theorem showing that hypotraceable graphs possessing some connectivity properties are all built using either Thomassen's or our method. We also prove that if G is a Grinbergian graph without a triangular region, then G is not maximal nonhamiltonian and using the proof method we construct a hypohamiltonian graph of order 36 with crossing number 1, improving the best‐known bound of 46 [14].     

On graphs whose square have strong hamiltonian properties

The squareG² of a graph G is the graph having the same vertex set as G and two vertices are adjacent if and only if they are at distance at most 2 from each other. It is known that if G has no cut-vertex, then G² is Hamilton-connected (see [G. Chartrand, A.M. Hobbs, H.A. Jung, S.F. Kapoor, C.St.J.A. Nash-Williams, The square of a block is hamiltonian connected, J. Combin. Theory Ser. B 16 (1974) 290-292; R.J. Faudree and R.H. Schelp, The square of a block is strongly path connected, J. Combin. Theory Ser. B 20 (1976) 47-61]). We prove that if G has only one cut-vertex, then G² is Hamilton-connected. In the case that G has only two cut-vertices, we prove that if the block that contains the two cut-vertices is hamiltonian, then G² is Hamilton-connected. Further, we characterize all graphs with at most one cycle having Hamilton-connected square.     

Hypohamiltonian Planar Cubic Graphs with Girth 5

A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4‐cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity 5, and thus girth 5. We show by computer search that the smallest members of this class are three graphs with 76 vertices.     

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

On s‐Hamiltonian Line Graphs

For an integer s ≥ 0, a graph G is s‐hamiltonian if for any vertex subset with |S^′| ≤ s, G ‐ S^′ is hamiltonian. It is well known that if a graph G is s‐hamiltonian, then G must be (s+2)‐connected. The converse is not true, as there exist arbitrarily highly connected nonhamiltonian graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s‐hamiltonian if and only if it is (s+2)‐connected.     

On semisymmetric cubic graphs of order 6p~2

A graph r is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut(Г) acting transitively on its edge set but not on its vertex set. In the case of G. = A, we call r a semisymmetric graph. The aim of this paper is to investigate (G-)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6p2 for an odd prime p.     

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.     

Hamiltonicity and pancyclicity of cartesian products of graphs

The cartesian product of a graph G with K₂ is called a prism over G. We extend known conditions for hamiltonicity and pancyclicity of the prism over a graph G to the cartesian product of G with paths, cycles, cliques and general graphs. In particular we give results involving cubic graphs and almost claw-free graphs.We also prove the following: Let G and H be two connected graphs. Let both G and H have a 2-factor. If Δ(G)≤g^′(H) and Δ(H)≤g^′(G) (we denote by g^′(F) the length of a shortest cycle in a 2-factor of a graph F taken over all 2-factorization of F), then G□H is hamiltonian.     

Cubic vertex‐transitive graphs of order 2pq

A graph is vertex?transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1?S and S={s^?1 | s∈S}. The Cayleygraph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g∈G, s∈S}. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4p or 2p² and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

The rank-width of the square grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if, and only if, it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width.The n×nsquare grid, denoted by G_n,n, is a graph on the vertex set {1,2,…,n}×{1,2,…,n}, where a vertex (x,y) is connected by an edge to a vertex (x^′,y^′) if and only if |x−x^′|+|y−y^′|=1.We prove that the rank-width of G_n,n is equal to n−1, thus solving an open problem of Oum.     

One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.     

Series parallel extensions of plane graphs to dual-eulerian graphs

A plane graph is dual-eulerian if it has an eulerian tour with the property that the same sequence of edges also forms an eulerian tour in the dual graph. Dual-eulerian graphs are of interest in the design of CMOS VLSI circuits.Every dual-eulerian plane graph also has an eulerian Petrie (left–right) tour thus we consider series-parallel extensions of plane graphs to graphs, which have eulerian Petrie tours. We reduce several special cases of extensions to the problem of finding hamiltonian cycles. In particular, a 2-connected plane graph G has a single series parallel extension to a graph with an eulerian Petrie tour if and only if its medial graph has a hamiltonian cycle.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

On large light graphs in families of polyhedral graphs

A graph H is said to be light in a family H of graphs if each graph G∈H containing a subgraph isomorphic to H contains also an isomorphic copy of H such that each its vertex has the degree (in G) bounded above by a finite number φ(H,H) depending only on H and H. We prove that in the family of all 3-connected plane graphs of minimum degree 5 (or minimum face size 5, respectively), the paths with certain small graphs attached to one of its ends are light.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Every 4-Connected Line Graph of a Planar Graph is Hamiltonian

Let G be a graph withE(G) $#x2260;ø. The line graph of G, written L(G) hasE(G) as its vertex set, where two vertices are adjacent in L(G) if and only if the corresponding edges are adjacent inG. Thomassen conjectured that all 4-connected line graphs are hamiltonian [2]. We show that this conjecture holds for planar graphs.     

Forcing faces in plane bipartite graphs

Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G∈Ω is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.