On geodesic structures of weakly median graphs I. Decomposition and octahedral graphs

We prove that the non-trivial (finite or infinite) weakly median graphs which are undecomposable with respect to gated amalgamation and Cartesian multiplication are the 5-wheels, the subhyperoctahedra different from K₁, the path K_1,2 and the 4-cycle K_2,2, and the two-connected K₄- and K_1,1,3-free bridged graphs. These prime graphs are exactly the weakly median graphs which do not have any proper gated subgraphs other than singletons. For finite graphs, these results were already proved in [H.-J. Bandelt, V.C. Chepoi, The algebra of metric betweenness I: subdirect representation, retracts, and axiomatics of weakly median graphs, preprint, 2002]. A graph G is said to have the half-space copoint property (HSCP) if every non-trivial half-space of the geodesic convexity of G is a copoint at each of its neighbors. It turns out that any median graph has the HSCP. We characterize the weakly median graphs having the HSCP. We prove that the class of these graphs is closed under gated amalgamation and Cartesian multiplication, and we describe the prime and the finite regular elements of this class.     

The graphs with only self-dual signings

Given a graph G, it is possible to attach positive and negative signs to its lines only, to its points only, or to both. The resulting structures are called respectively signed graphs, marked graphs and nets. The dual of each such structure is obtained by changing every sign in it. We determine all graphs G for which every suitable marked graph on G is self-dual (the M-dual graphs), and also the corresponding graphs G for signed graphs (S-dual) and for nets (N-dual.A graph G is M-dual if and only if G or ? is one of the graphs K_2m, 2K_m, mK₂, K_m + K₂ or 2C₄. The S-dual graphs are C₆, 2C₃, 2C₄, 2K_1n, 2nK₂, K_1,2n, nK_1,2, K_2n, K?_n and all graphs obtained from these by the addition of isolated points. Finally, the only N-dual graph other than -K_2n is 2K₂.     

Ear-decompositions of matching-covered graphs

We call a graphmatching-covered if every line belongs to a perfect matching. We study the technique of “ear-decompositions” of such graphs. We prove that a non-bipartite matching-covered graph containsK ₄ orK ₂⊕K ₃ (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of τ-critical graphs.     

Kuratowski-Pontrjagin theorem on planar graphs

In this paper we give a new proof of the well-known theorem of Kuratowski-Pontrjagin on planar graphs: A graph is planar iff it does not contain a subgraph homeomorphic to K₅ or K_3,3.     

On choosability of some complete multipartite graphs and Ohba's conjecture

A graph G is said to be chromatic-choosable if ch(G)=χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. But for complete multipartite graphs, the graphs for which Ohba's conjecture has been verified are nothing more than K_{3*2,2*(k-3),1}, K_3,2*(k-1), and K_{s+3,2*(k-s-1),1*s}. These results have been obtained indirectly from the investigation about complete multipartite graphs by Gravier and Maffray and by Enomoto et al. In this paper we show that Ohba's conjecture is true for complete multipartite graphs K_{4,3,2*(k-4),1*2} and K_{5,3,2*(k-5),1*3}. By the way, we give some discussions about a result of Enomoto et al.     

Some families of integral graphs

A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs K_1,r•K_n, r∗K_n, K_1,r•K_m,n, r∗K_m,n and the tree K_1,s•T(q,r,m,t) are defined. We determine the characteristic polynomials of these graphs and also obtain sufficient and necessary conditions for these graphs to be integral. Some sufficient conditions are found by using the number theory and computer search. All these classes are infinite. Some new results which treat interrelations between integral trees of various diameters are also found. The discovery of these integral graphs is a new contribution to the search of such graphs.     

Characterizations for co-graphs defined by restricted NLC-width or clique-width operations

In this paper we characterize subclasses of co-graphs defined by restricted NLC-width operations and subclasses of co-graphs defined by restricted clique-width operations.We show that a graph has NLCT-width 1 if and only if it is (C₄,P₄)-free. Since (C₄,P₄)-free graphs are exactly trivially perfect graphs, the set of graphs of NLCT-width 1 is equal to the set of trivially perfect graphs, and a recursive definition for trivially perfect graphs follows. Further we show that a graph has linear NLC-width 1 if and only if is (C₄,P₄,2K₂)-free. This implies that the set of graphs of linear NLC-width 1 is equal to the set of threshold graphs.We also give forbidden induced subgraph characterizations for co-graphs defined by restricted clique-width operations using P₄, 2K₂, and co-2P₃.     

On graphs whose star sets are (co-)cliques

In this paper we study graphs all of whose star sets induce cliques or co-cliques. We show that the star sets of every tree for each eigenvalue are independent sets. Among other results it is shown that each star set of a connected graph G with three distinct eigenvalues induces a clique if and only if G=K_1,2 or K_2,…,2. It is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets.     

On Connected 2-K n -Residual Graphs

A graph G is said to be K _n -residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to K _n . We inductively define a multiply-K _n -residual graph by saying that G is m-K _n -residual if the removal of the closed neighborhood of any vertex of G results in an (m – 1)-K _n -residual graphs. Erdös, Harary and Klawe [2] determined the minimum order of the m?K _n -residual graphs for all m and n, which are not necessarily connected, the minimum order of connected; K _n -residual graphs, all K _n -residual extremal graphs. They also stated some conjectures regarding the connected case. In this paper, we determine the minimum order of a connected 2-K _n -residual graph and specify the extremal graphs, expect for n = 3. In particular, we determining only one connected 2-K ₄-residual graph of minimal order, and show that there is a connected 2-K ₆-residual graph non isomorphic to K ₈ × K ₃ with minimum order. Finally we present and a revised version of the conjecture in [2].     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

Bounds for the b-chromatic number of some families of graphs

In this paper we obtain some upper bounds for the b-chromatic number of K_1,s-free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are given in terms of either the clique number or the chromatic number of a graph or the biclique number for a bipartite graph. We show that all the bounds are tight.     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

Spanning cubic graph designs

Graph designs are natural extensions of BIBDs (balanced incomplete block designs). In this paper we explore spanning cubic graph designs and develop tools for constructing some of them. We show that K₁₆ can be decomposed into each of the 4060 connected cubic graphs of order 16, and into precisely 144 of the 147 disconnected cubic graphs of order 16. We also identify some infinite families of cubic graphs of order 6n+4 that decompose K_6n+4.     

A characterization of complete matroid base graphs

This study grew from an attempt to give a local analysis of matroid base graphs. A neighborhood-preserving covering of graphs p:G → H is one such that p restricted to every neighborhood in G is an isomorphism. This concept arises naturally when considering graphs with a prescribed set of local properties. A characterization is given of all connected graphs with two local properties: (a) there is a pair of adjacent points, the intersection of whose neighborhoods does not contain three mutually nonadjacent points; (b) the intersection of the neigh-borhoods of points two apart is a 4-cycle. Such graphs have neighborhoods of the form K_n × K_m for fixed n, m and are either complete matroid base graphs or are their images under neighborhood-preserving coverings. If n ≠ m, the graph is unique; if n = m, there are n ? 3 such images which are nontrivial. These examples prove that no set of properties of bounded diameter can characterize matroid base graphs.     

2K2 vertex-set partition into nonempty parts

A graph is 2K₂-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C, D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. Determining whether an arbitrary graph is 2K₂-partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We show that for C₄-free graphs, circular-arc graphs, spiders, P₄-sparse graphs, and bipartite graphs the 2K₂-partition problem can be solved in polynomial time.     

On the 2-rainbow domination in graphs

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G□K₂. In this paper, we show that the problem of deciding if a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs. Exact values of 2-rainbow domination numbers of several classes of graphs are found, and it is shown that for the generalized Petersen graphs GP(n,k) this number is between ⌈4n/5⌉ and n with both bounds being sharp.     

Turán numbers for K s,t -free graphs: Topological obstructions and algebraic constructions

We show that every hypersurface in ? ^s × ? ^s contains a large grid, i.e., the set of the form S × T, with S, T ? ? ^s . We use this to deduce that the known constructions of extremal K _2,2-free and K _3,3-free graphs cannot be generalized to a similar construction of K _s,s -free graphs for any s ≥ 4. We also give new constructions of extremal K _s,t -free graphs for large t.     

Block designs and graph imbeddings

Heffter first observed that certain imbeddings of complete graphs give rise to BIBD's with k = 3 and λ = 2 (and sometimes λ = 1); Alpert established a one-to-one correspondence between BIBD's with k = 3 and λ = 2 and triangulation systems for complete graphs. In this paper we extend this correspondence to PBIBD's on two association classes with k = 3, λ₁ = 0 and λ₂ = 2, and triangulation systems for strongly regular graphs. The group divisible designs of Hanani are used to construct triangulations for the graphs K_n(m), in each case permitted by the euler formula. Conversely, triangular imbeddings of K_n(m) are constructed which lead to new group divisible designs. A process is developed for “doubling” a given PBIBD of an appropriate form. Various extensions of these ideas are discussed, as is an application to the construction of quasigroups.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.     

Graphs with bounded valency that do not embed in the projective plane

In 1930 Kuratowski proved that a graph does not embed in the real plane R² if and only if it contains a subgraph homeomorphic to one of two graphs, K₅ or K₃₃. Let I_n(P) denote the minimal set of graphs whose vertices have miximal valency n such that any graph which does not embed in the real projective plane (or equivalently, does not embed in the Möbius band) contains a subgraph homeomorphic to a graph in I_n(P) for some positive integer n. Glover and Huneke and Milgram proved that there are only 6 graphs in I₃(P). This note proves that for each n, I_n(P) is finite.