On Beck’s coloring of posets

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset P with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of P with two vertices x,y adjacent if 0 is the only element lying below x and y. We prove that for such graphs, the chromatic number and the clique number coincide. Also, we give a condition under which posets are not finitely colorable.     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

On Strongly Regular Graphs with k=2μ and Their Extensions

We obtain a convenient expression for the parameters of a strongly regular graph with k=2 in terms of the nonprincipal eigenvalues x and –y. It turns out in particular that such graphs are pseudogeometric for pG _x(2x,y–1). We prove that a strongly regular graph with parameters (35,16,6,8) is a quotient of the Johnson graph (8,4). We also find the parameters of strongly regular graphs in which the neighborhoods of vertices are pseudogeometric graphs for pG _x(2x,t),x3. In consequence, we establish that a connected graph in which the neighborhoods of vertices are pseudogeometric graphs for pG ₃(6,2) is isomorphic to the Taylor graph on 72 vertices or to the alternating form graph Alt(4,2) with parameters (64,35,18,20).     

Sur Les Partitions en Chaînes du Graphe Divisoriel

On path partitions of the divisor graph. Let D(x) be the graph with vertices {1, 2, ..., ⌊x⌋} whose edges come from the division relation, and let D(x, y) be the subgraph restricted to the integers with prime factors less than or equal to y. We give sufficient conditions on x and y for the graph D(x, y) to be Hamiltonian. We deduce an asymptotic formula for the number of paths in D(x) needed to partition the set of vertices {1, 2, ..., ⌊x⌋}.

     

The domination and competition graphs of a tournament

Vertices x and y dominate a tournament T if for all vertices z ≠ x, y, either x beats z or y beats z. Let dom(T) be the graph on the vertices of T with edges between pairs of vertices that dominate T. We show that dom(T) is either an odd cycle with possible pendant vertices or a forest of caterpillars. While this is not a characterization, it does lead to considerable information about dom(T). Since dom(T) is the complement of the competition graph of the tournament formed by reversing the arcs of T, complementary results are obtained for the competition graph of a tournament. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 103–110, 1998     

Domination Graphs with Nontrivial Components

 A tournament is an oriented complete graph. Vertices x and y dominate a tournament T if for all vertices z≠x,y, either (x,z) or (y,z) are arcs in T (possibly both). The domination graph of a tournament T is the graph on the vertex set of T containing edge {x,y} if and only if x and y dominate T. In this paper we determine which graphs containing no isolated vertices are domination graphs of tournaments. Received: May 20, 1998 Final version received: May 26, 1999     

Clique or hole in claw-free graphs

Given a claw-free graph and two non-adjacent vertices x and y without common neighbours we prove that there exists a hole through x and y unless the graph contains the obvious obstruction, namely a clique separating x and y. We derive two applications: We give a necessary and sufficient condition for the existence of an induced x-z path through y, where x,y,z are prescribed vertices in a claw-free graph; and we prove an induced version of Menger?s theorem between four terminal vertices. Finally, we improve the running time for detecting a hole through x and y and for the Three-in-a-Tree problem, if the input graph is claw-free.     

Cube intersection concepts in median graphs

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k≥0, the intersection graph Q_k(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices H_x and H_y in Q_k(G) are adjacent whenever the intersection H_x∩H_y contains a subgraph isomorphic to Q_k. Characterizations of clique-graphs in terms of these intersection concepts when k>0, are presented. Furthermore, we introduce the so-called maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted , whose vertices are maximal hypercubes of G, and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to . We also study convergence of median graphs to the one-vertex graph with respect to all these operations.     

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)?{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ _I (R), with vertices {x ∈ R?I | xy ∈ I for some y ∈ R?I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ _I (R). We also determine when Γ _I (R) is either a complete graph or a complete bipartite graph and investigate when Γ _I (R) ? Γ(S) for some commutative ring S.     

On constructible graphs,locally Helly graphs,and convexity

A (finite or infinite) graph G is constructible if there exists a well‐ordering ≤ of its vertices such that for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x. Particular constructible graphs are Helly graphs and connected bridged graphs. In this paper we study a new class of constructible graphs, the class of locally Helly graphs. A graph G is locally Helly if, for every pair (x,y) of vertices of G whose distance is d ≥ 2, there exists a vertex whose distance to x is d ? 1 and which is adjacent to y and to all neighbors of y whose distance to x is at most d. Helly graphs are locally Helly, and the converse holds for finite graphs. Among different properties we prove that a locally Helly graph is strongly dismantable, hence cop‐win, if and only if it contains no isometric rays. We show that a locally Helly graph G is finitely Helly, that is, every finite family of pairwise non‐disjoint balls of G has a non‐empty intersection. We give a sufficient condition by forbidden subgraphs so that the three concepts of Helly graphs, of locally Helly graphs and of finitely Helly graphs are equivalent. Finally, generalizing different results, in particular those of Bandelt and Chepoi 1 about Helly graphs and bridged graphs, we prove that the Helly number h(G) of the geodesic convexity in a constructible graph G is equal to its clique number ω(G), provided that ω(G) is finite. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 280–298, 2003     

When is the Comaximal Graph Split?

Let R be a commutative ring with non zero unity. Let Ω(R) be a graph with vertices as elements of R whose two distinct vertices x and y are adjacent if and only if Rx + Ry = R. A graph (V, E) is said to be a split graph if V is the disjoint union of two sets K and S where K induces a complete subgraph and S is an independent set. We investigate the properties of R when Ω(R) is split.     

Realization of digraphs by preferences based on distances in graphs

Consider a graph G with two distinguished sets of vertices: the voters and the candidates. A voter v prefers candidate x to candidate y if d(v, x) < d(v, y). This preference relation defines an asymmetric digraph whose vertices are the candidates, in which there is an arc from candidate x to candidate y if and only if more voters prefer x to y than prefer y to x. T. W. Johnson and P. J. Slater (“Realization of Majority Preference Digraphs by Graphically Determined Voting Patterns,” Congressus Numerantium, vol. 67 [1988] 175-186) have shown that each asymmetric digraph of order n can be realized in this way using a graph of order O(n²). We present a new construction reducing the quadratic upper bound to a linear bound. © 1995 John Wiley & Sons, Inc.     

Restricted Fault Diameter of Hypercube Networks

This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n ≥ 2).It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2n-3 vertices in Qn - {x, y}, if F contains neither of neighbor-sets of x and y in Qn, then the distance between x andy in Qn - F is given by D(Qn-F;x,y){=1 , for=1;≤d 4 , for 2≤d≤n-2,n≥4;≤n 1, for d=n-1,n≥3; =n, for d=n. Furthermore, the upper bounds are tight. As an immediately consequence, Qn can tolerate up to 2n-3 vertices failures and remain diameter 4 if n = 3 and n 2 if n ≥ 4 provided that for each vertex x in Qn, all the neighbors of x do not fail at the same time. This improves Esfahanian‘s result.     

Circular colorings of weighted graphs

Suppose that G is a finite simple graph and w is a weight function which assigns to each vertex of G a nonnegative real number. Let C be a circle of length t. A t-circular coloring of (G, w) is a mapping Δ of the vertices of G to arcs of C such that Δ(x)∩Δ(y) = 0 if (x, y) ∈ E(G) and Δ(x) has length w(x). The circular-chromatic number of (G, w) is the least t for which there is a t-circular coloring of (G, w). This paper discusses basic properties of circular chromatic number of a weighted graph and relations between this parameter and other graph parameters. We are particularly interested in graphs G for which the circular-chromatic number of (G, w) is equal to the fractional clique weight of (G, w) for arbitrary weight function w. We call such graphs star-superperfect. We prove that odd cycles and their complements are star-superperfect. We then prove a theorem about the circular chromatic number of lexicographic product of graphs which provides a tool of constructing new star-superperfect graphs from old ones. © 1996 John Wiley & Sons, Inc.     

The Local Structure of a Bipartite Distance-regular Graph

In this paper, we consider a bipartite distance-regular graph Γ = (X, E) with diameter d ≥ 3. We investigate the local structure ofΓ , focusing on those vertices with distance at most 2 from a given vertex x. To do this, we consider a subalgebra R = R(x) ofMat_{0307a0x.gif X}(C), where 0307a1x.gifX denotes the set of vertices in X at distance 2 from x. R is generated by matrices Ã, 0307a2x.gif J, and 0307a3x.gif D defined as follows. For all y, z 0307a4x.gif X, the (y,z )-entry of à is 1 if y, z are at distance 2, and 0 otherwise. The (y, z)-entry of 0307a5x.gif J equals 1, and the (y,z )-entry of 0307a6x.gif D equals the number of vertices of X adjacent to each ofx , y, and z. We show that R is commutative and semisimple, with dimension at least 2. We assume that R is one of 2, 3, or 4, and explore the combinatorial implications of this. We are motivated by the fact that if Γ has a Q-polynomial structure, then R ≤ 4.     

The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

Geodesics in Graphs,an Extremal Set Problem,and Perfect Hash Families

 A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality of a collection 𝒮 of subsets of [n]={1,2,…,n} such that, for any two distinct elements x,y∈[n], there exists disjoint subsets A _x ,A _y ∈𝒮 such that x∈A _x and y∈A _y . This separating set problem can be generalized, and some bounds can be obtained from known results on families of hash functions. Received: May 19, 2000 Final version received: July 5, 2001     

Hamiltonicity of 3-Arc Graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. We prove that any connected 3-arc graph is hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are hamiltonian. As a corollary we obtain that any vertex-transitive graph which is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three must be hamiltonian. This confirms the conjecture, for this family of vertex-transitive graphs, that all vertex-transitive graphs with finitely many exceptions are hamiltonian. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.     

Very Asymmetric Marking Games

We investigate a competitive version of the coloring number of a graph G = (V, E). For a fixed linear ordering L of V let s (L) be one more than the maximum outdegree of G when G is oriented so that x ← y if x < _L y. The coloring number of G is the minimum of s (L) over all such orderings. The (a, b)-marking game is played on a graph G = (V, E) as follows. At the start all vertices are unmarked. The players, Alice and Bob, take turns playing. A play consists of Alice marking a unmarked vertices or Bob marking b unmarked vertices. The game ends when there are no remaining unmarked vertices. Together the players create a linear ordering L of V defined by x < y if x is marked before y. The score of the game is s (L). The (a, b)-game coloring number of G is the minimum score that Alice can obtain regardless of Bob’s strategy. The usual (1, 1)-marking game is well studied and there are many interesting results. Our main result is that if G has an orientation with maximum outdegree k then the (k, 1)-game coloring number of G is at most 2k + 2. This extends a fundamental result on the (1, 1)-game coloring number of trees. We also construct examples to show that this bound is tight for many classes of graphs. Finally we prove bounds on the (a, 1)-game coloring number when a < k.