Subdivisions and the chromatic index of r-graphs

Let T₂ be the graph obtained from the Petersen graph by first deleting a vertex and then contracting an edge incident to a vertex of degree two. We give a simple characterization of the graphs that contain no subdivision of T₂. This characterization is used to show that if every planar r-graph is r-edge colorable, then every r-graph with no subdivision of T₂ is r-edge colorable. © 1996 John Wiley & Sons, Inc.     

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.     

Forbidden subgraphs and hamiitonian properties in the square of a connected graph

Various Harniltonian-like properties are investigated in the squares of connected graphs free of some set of forbidden subgraphs. The star K_1,4 the subdivision graph of K_1,3, and the subdivision graph of K_1,3 minus an endvertex play central roles. In particular, we show that connected graphs free of the subdivision graph of K_1,3 minus an endvertex have vertex pancyclic squares.     

Excluding Induced Subdivisions of the Bull and Related Graphs

For any graph H, let Forb*(H) be the class of graphs with no induced subdivision of H. It was conjectured in [J Graph Theory, 24 (1997), 297–311] that, for every graph H, there is a function f_H: ?→? such that for every graph G∈Forb*(H), χ(G)≤f_H(ω(G)). We prove this conjecture for several graphs H, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex‐disjoint pendant edges), and what we call a “necklace,” that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:49–68, 2012     

Nowhere‐zero 3‐flows in locally connected graphs

Let G be a graph. For each vertex v ∈V(G), N_v denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k‐edge‐connected if for each vertex v ∈V(G), N_v is k‐edge‐connected. In this paper we study the existence of nowhere‐zero 3‐flows in locally k‐edge‐connected graphs. In particular, we show that every 2‐edge‐connected, locally 3‐edge‐connected graph admits a nowhere‐zero 3‐flow. This result is best possible in the sense that there exists an infinite family of 2‐edge‐connected, locally 2‐edge‐connected graphs each of which does not have a 3‐NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211–219, 2003     

A mixed 0-1 linear programming formulation for the exact solution of the minimum linear arrangement problem

We are concerned with the exact solution of a graph optimization problem known as minimum linear arrangement (MinLA). Define the length of each edge of a graph with respect to a linear ordering of the graph vertices. Then, the MinLA problem asks for a vertex ordering that minimizes the sum of edge lengths. MinLA has several practical applications and is NP-Hard. We present a mixed 0-1 linear programming formulation of the problem, which led to fast optimal solutions for dense graphs of sizes up to n = 23.     

Super-Connectivity and Hyper-Connectivity of Vertex Transitive Bipartite Graphs

A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if the deletion of each minimum vertex cut creates exactly two components, one of which is an isolated vertex. In this note, we proved that a vertex transitive bipartite graph is not super-connected if and only if it is isomorphic to the lexicographic product of a cycle C_n(n ≥ 6) by a null graph N_m. We also characterized non-hyper-connected vertex transitive bipartite graphs.     

On the facets of the lift-and-project relaxations of graph subdivisions

We study the behavior of lift-and-project procedures for solving combinatorial optimization problems as described by Lovász and Schrijver (1991), in the context of the stable set problem on graphs. Following the work of Wolsey (1976), we investigate how to generate facets of the relaxations obtained by these procedures from facets of the relaxations of the original graph, after applying fundamental graph operations. We show our findings for the odd subdivision of an edge and its generalization, the stretching of a vertex operation.     

On triangles in ‐minor free graphs

We study graphs where each edge that is incident to a vertex of small degree (of degree at most 7 and 9, respectively) belongs to many triangles (at least 4 and 5, respectively) and show that these graphs contain a complete graph (K₆ and K₇, respectively) as a minor. The second case settles a problem of Nevo. Moreover, if each edge of a graph belongs to six triangles, then the graph contains a K₈‐minor or contains K_{2, 2, 2, 2, 2} as an induced subgraph. We then show applications of these structural properties to stress freeness and coloring of graphs. In particular, motivated by Hadwiger's conjecture, we prove that every K₇‐minor free graph is 8‐colorable and every K₈‐minor free graph is 10‐colorable.     

Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity

In this work we show that among all n-vertex graphs with edge or vertex connectivity k, the graph G=K_k(K₁+K_n−k−1), the join of K_k, the complete graph on k vertices, with the disjoint union of K₁ and K_n−k−1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum.     

splitting number is NP-complete

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k⩾0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v₁ and v₂, and attaches the neighbors of v either to v₁ or to v₂. We prove that the splitting number decision problem is NP-complete when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs implies NP-completeness for graphs not containing a subdivision of K₅ as a subgraph.     

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.     

Super-connected edge transitive graphs

A graph G is said to be super-connected if any minimum cut of G isolates a vertex. In a previous work due to the second author of this note, super-connected graphs which are both vertex transitive and edge transitive are characterized. In this note, we generalize the characterization to edge transitive graphs which are not necessarily vertex transitive, showing that the only irreducible edge transitive graphs which are not super-connected are the cycles C_n(n?6) and the line graph of the 3-cube, where irreducible means the graph has no vertices with the same neighbor set. Furthermore, we give some sufficient conditions for reducible edge transitive graphs to be super-connected.     

The connectivity of the block-intersection graphs of designs

It is shown that the vertex connectivity of the block-intersection graph of a balanced incomplete block design,BIBD (v, k, 1), is equal to its minimum degree. A similar statement is proved for the edge connectivity of the block-intersection graph of a pairwise balanced design,PBD (v, K, 1). A partial result on the vertex connectivity of these graphs is also given. Minimal vertex and edge cuts for the corresponding graphs are characterized.Research supported in part by a B.C. Science Council G.R.E.A.T. Scholarship.Research supported in part by an NSERC Postdoctoral Fellowship.     

Roman domination subdivision number of graphs

A Roman dominating function on a graph G = (V, E) is a function f : V ? {0, 1, 2}f : V \rightarrow \{0, 1, 2\} satisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function is the value w(f) = ?_{v ? V} f(v)w(f) = \sum_{v\in V} f(v). The Roman domination number of a graph G, denoted by _gR(G)_{\gamma R}(G), equals the minimum weight of a Roman dominating function on G. The Roman domination subdivision number sd_gR(G)sd_{\gamma R}(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number. In this paper, first we establish upper bounds on the Roman domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that $1 \leq sd_{\gamma R}(G) \leq 3$1 \leq sd_{\gamma R}(G) \leq 3. Finally, we show that the Roman domination subdivision number of a graph can be arbitrarily large.     

Copositive programming motivated bounds on the stability and the chromatic numbers

The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovász theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs. Partial support by the EU project Algorithmic Discrete Optimization (ADONET), MRTN-CT-2003-504438, is gratefully acknowledged.     

Upper bounds on vertex distinguishing chromatic index of some Halin graphs

A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by ???? _s (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with ??(G) ?? 4, respectively.     

On the commutativity of antiblocker diagrams under lift-and-project operators

The behavior of the disjunctive operator, defined by Balas, Ceria and Cornuéjols, in the context of the “antiblocker duality diagram” associated with the stable set polytope, QSTAB(G), of a graph and its complement, was first studied by Aguilera, Escalante and Nasini. The authors prove the commutativity of this diagram in any number of iterations of the disjunctive operator. One of the main consequences of this result is a generalization of the Perfect Graph Theorem under the disjunctive rank.In the same context, Lipták and Tunçel study the lift-and-project operators N₀, N and N₊ defined by Lovász and Schrijver. They find a graph for which the diagram does not commute in one iteration of the N₀- and N-operator. In connection with N₊, the authors implicitly suggest a similar result proving that if the diagram commutes in k=O(1) iterations, P=NP.In this paper, we give for any number of iterations, explicit proofs of the non commutativity of the N₀-, N- and N₊-diagram.In the particular case of the N₀- and N-operator, we find bounds for the ranks of the complements of line graphs (of complete graphs), which allow us to prove that the diagrams do not commute for these graphs.     

New edge neighborhood graphs

Let G be an undirected simple connected graph, and e = uv be an edge of G. Let N _G(e) be the subgraph of G induced by the set of all vertices of G which are not incident to e but are adjacent to u or v. Let N _e be the class of all graphs H such that, for some graph G,N _G (e) H for every edge e of G. Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in N _e. Balasubramanian and Alsardary [1] obtained some other graphs in N _e. In this paper we given some new graphs in N _e.     

Ordered interval routing schemes

An Interval Routing Scheme (IRS) represents the routing tables in a network in a space-efficient way by labeling each vertex with an unique integer address, and the outgoing edges at each vertex with disjoint subintervals of these addresses. An IRS that has at most k intervals per edge label is called a k-IRS. In this paper, we propose a new type of interval routing scheme, called an Ordered Interval Routing Scheme (OIRS), that uses an ordering of the outgoing edges at each vertex and allows non-disjoint intervals in the labels of those edges. We show for a number of graph classes that using an OIRS instead of an IRS reduces the size of the routing tables in the case of optimal routing, i.e., routing along shortest paths. We show that optimal routing in any k-tree is possible using an OIRS with at most 2^k−1 intervals per edge label, although the best known result for an IRS is 2^k+1 intervals per edge label. Any torus has an optimal 1-OIRS, although it may not have an optimal 1-IRS. We present similar results for the Petersen graph, k-garland graphs and a few other graphs.