Isometric embeddings of subdivided connected graphs into hypercubes

Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollack [R.L. Graham, H. Pollack, On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495-2519; and D. Djokovi?, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263-267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specifically, we deal with the case where this configuration is a connected graph of order 4, a complete graph of order 5 and the case of a k-fan F_k(k≥3).     

On convex embeddings of planar 3‐connected graphs

A well‐known Tutte's theorem claims that every 3‐connected planar graph has a convex embedding into the plane. Tutte's arguments also show that, moreover, for every nonseparating cycle C of a 3‐connected graph G, there exists a convex embedding of G such that C is a boundary of the outer face in this embedding. We give a simple proof of this last result. Our proof is based on the fact that a 3‐connected graph admits an ear assembly having some special properties with respect to the nonseparating cycles of the graph. This fact may be interesting and useful in itself. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 120–124, 2000     

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.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes

We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Δ can be coloured with at most ?(Δ)=MΔ²⁶$?(\mathrm{\Delta })=M{\mathrm{\Delta }}^{26}$ colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most ?(Δ)$?(\mathrm{\Delta })$ trees, and that the event structure whose domain is X admits a nice labelling with ?(Δ)$?(\mathrm{\Delta })$ labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.     

Note on non-separating and removable cycles in highly connected graphs

We combine two well-known results by Mader and Thomassen, respectively. Namely, we prove that for any k-connected graph G (k≥4), there is an induced cycle C such that G−V(C) is (k−3)-connected and G−E(C) is (k−2)-connected. Both “(k−3)-connected” and “(k−2)-connected” are best possible in a sense.     

On double bound graphs and forbidden subposets

For a poset P=(X,≤_P), the double bound graph (DB-graph) of P is the graph DB(P)=(X,E_DB(P)), where xy∈E_DB(P) if and only if x≠y and there exist n,m∈X such that n≤_Px,y≤_Pm. We obtain that for a subposet Q of a poset P,Q is an (n, m)-subposet of P if and only if DB(Q) is an induced subgraph DB(P). Using this result, we show some characterizations of split double bound graphs, threshold double bound graphs and difference double bound graphs in terms of (n, m)-subposets and double canonical posets.     

On the number of triangular embeddings of complete graphs and complete tripartite graphs

We prove that for every prime number p and odd m>1, as s→∞, there are at least w face 2‐colorable triangular embeddings of K_{w, w, w}, where w = m·p^s. For both orientable and nonorientable embeddings, this result implies that for infinitely many infinite families of z, there is a constant c>0 for which there are at least z nonisomorphic face 2‐colorable triangular embeddings of K_z. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Light subgraphs in planar graphs of minimum degree 4 and edge‐degree 9

A graph H is light in a given class of graphs if there is a constant w such that every graph of the class which has a subgraph isomorphic to H also has a subgraph isomorphic to H whose sum of degrees in G is ≤ w. Let be the class of simple planar graphs of minimum degree ≥ 4 in which no two vertices of degree 4 are adjacent. We denote the minimum such w by w(H). It is proved that the cycle C_s is light if and only if 3 ≤ s ≤ 6, where w(C₃) = 21 and w(C₄) ≤ 35. The 4‐cycle with one diagonal is not light in , but it is light in the subclass consisting of all triangulations. The star K_1,s is light if and only if s ≤ 4. In particular, w(K_1,3) = 23. The paths P_s are light for 1 ≤ s ≤ 6, and heavy for s ≥ 8. Moreover, w(P₃) = 17 and w(P₄) = 23. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 261–295, 2003     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n^0.694), and the circumference of a 3-connected claw-free graph is Ω(n^0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m^0.753) edges. We use this result together with the Ryjá?ek closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n^0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs.     

无爪图中不相邻子图P_4和K_1的度和条件下的哈密尔顿性

研究子图的度和图的哈密尔顿性的关系,证明图~$G$ 是一个~$n$ 阶~3-\,连通无爪图且最小度~$\delta(G)\geq4$, 如果图~$G$ 中任意两个分别同构于~$P_4$, $K_1$ 的不相邻子图~$H_1$, $H_2$ 满足~$d(H_1)+d(H_2)\geq n$, 则图~$G$ 是哈密尔顿连通.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

Nearly light cycles in embedded graphs and crossing‐critical graphs

We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k‐crossing‐critical graph has crossing number at most 2k + 23. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 151–156, 2006     

Exponentially many maximum genus embeddings and genus embeddings for complete graphs

There are many results on the maximum genus, among which most are written for the existence of values of such embeddings, and few attention has been paid to the estimation of such embeddings and their applications. In this paper we study the number of maximum genus embeddings for a graph and find an exponential lower bound for such numbers. Our results show that in general case, a simple connected graph has exponentially many distinct maximum genus embeddings. In particular, a connected cubic graph G of order n always has at least distinct maximum genus embeddings, where α and m denote, respectively, the number of inner vertices and odd components of an optimal tree T. What surprise us most is that such two extremal embeddings (i.e., the maximum genus embeddings and the genus embeddings) are sometimes closely related with each other. In fact, as applications, we show that for a sufficient large natural number n, there are at least many genus embeddings for complete graph K _n with n ≡ 4, 7, 10 (mod12), where C is a constance depending on the value of n of residue 12. These results improve the bounds obtained by Korzhik and Voss and the methods used here are much simpler and straight. This work was supported by the National Natural Science Foundation of China (Grant No. 10671073), Science and Technology commission of Shanghai Municipality (Grant No. 07XD14011) and Shanghai Leading Academic Discipline Project (Grant No. B407)     

Short cycle structures for graphs on surfaces and an open problem of Mohar and Thomassen

In this paper we investigate cycle base structures of a (weighted) graph and show that much information of short cycles is contained in an MCB (i.e., minimum cycle base). After setting up a Hall type theorem for base-transformation, we give a sufficient and necessary condition for a cycle base to be an MCB. Furthermore, we show that the structure of MCB in a (weighted) graph is unique. The property is also true for those having a longest length (although much work has been down in evaluating MCB, little is known for those having a longest length). We use those methods to find out some unknown properties for short cycles sharing particular properties in (unweighted) graphs. As applications, we determine the structures of short cycles in an embedded graph and show that there exist polynomially bounded algorithms in finding a shortest contractible cycle and a shortest two-sided cycle provided such cycles exist. Those answer an open problem of B. Mohar and C. Thomassen.     

Netlike partial cubes, V: Completion and netlike classes

We define a completion of a netlike partial cube G by replacing each convex 2n-cycle C of G with n≥3 by an n-cube admitting C as an isometric cycle. We prove that a completion of G is a median graph if and only if G has the Median Cycle Property (MCP) (see N. Polat, Netlike partial cubes III. The Median Cycle Property, Discrete Math.). In fact any completion of a netlike partial cube having the MCP is defined by a universal property and turns out to be a minimal median graph containing G as an isometric subgraph. We show that the completions of the netlike partial cubes having the MCP preserves the principal constructions of these graphs, such as: netlike subgraphs, gated amalgams and expansions. Conversely any netlike partial cube having the MCP can be obtained from a median graph by deleting some particular maximal finite hypercubes. We also show that, given a netlike partial cube G having the MCP, the class of all netlike partial cubes having the MCP whose completions are isomorphic to those of G share different properties, such as: depth, lattice dimension, semicube graph and crossing graph.     

A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs

We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic triangular embeddings of the complete graph K_n is at least n^an2. A similar lower bound is also given, for an infinite set of values of n, on the number of nonisomorphic triangular embeddings of the complete regular tripartite graph K_n,n,n.     

Small subgraphs in random graphs and the power of multiple choices

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G₀ as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r−1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph GN.In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in GN for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N₀(F,r,n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1−o(1) as n→∞) avoids creating a copy of F for as long as N=o(N₀), and prove that any online strategy will a.a.s. create such a copy once N=ω(N₀).