Further results on the lower bounds of mean color numbers

Let G be a graph with n vertices. The mean color number of G, denoted by μ(G), is the average number of colors used in all n‐colorings of G. This paper proves that μ(G) ≥ μ(Q), where Q is any 2‐tree with n vertices and G is any graph whose vertex set has an ordering x₁,x₂,…,x_n such that x_i is contained in a K₃ of G[V_i] for i = 3,4,…,n, where V_i = {x₁,x₂,…,x_i}. This result improves two known results that μ(G) ≥ μ(O_n) where O_n is the empty graph with n vertices, and μ(G) ≥ μ(T) where T is a spanning tree of G. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 51–73, 2005     

The Shannon Capacity of a Union

For an undirected graph , let denote the graph whose vertex set is in which two distinct vertices and are adjacent iff for all i between 1 and n either or . The Shannon capacity c(G) of G is the limit , where is the maximum size of an independent set of vertices in . We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956. Received: December 8, 1997     

Privileged users in zero-error transmission over a noisy channel

The k-th power of a graph G is the graph whose vertex set is V(G) ^k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim _k→∞ α(G ^k )^1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = Σ _i C _i , describes a setting when there are t ≥ 2 senders, each with his own channel C _i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = Σ _i G _i . It is well known that c(G) ≥ Σ _i c(G _i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C _i to each sender i ∈ [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F ∈ F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k − 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp vertices contains all t colors. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research partially supported by a Charles Clore Foundation Fellowship.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

Note on a new coloring number of a graph

The distance coloring number X_d(G) of a graph G is the minimum number n such that every vertex of G can be assigned a natural number m ≤ n and no two vertices at distance i are both assigned i. It is proved that for any natural number n there exists a graph G with X_d(G) = n.     

Graph decompositions without isolated vertices III

Let G be a graph of order n, and n = Σ^k_i=1 a_i be a partition of n with a_i ≥ 2. In this article we show that if the minimum degree of G is at least 3k−2, then for any distinct k vertices v₁,…, v_k of G, the vertex set V(G) can be decomposed into k disjoint subsets A₁,…, A_k so that |A_i| = a_i,v_iisA_i is an element of A_i and “the subgraph induced by A_i contains no isolated vertices” for all i, 1 ≥ i ≥ k. Here, the bound on the minimum degree is sharp. © 1997 John Wiley & Sons, Inc.     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_{i = 1} a_i and σ₂(G) ≥ n + k − 1, then for any k vertices v₁, v₂,…, v_k in G, there exist vertex‐disjoint paths P₁, P₂,…, P_k such that |V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all a_i ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000     

Strong chromatic index of subset graphs

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S_m(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.     

Proof of a conjecture on game domination

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I₁×I₂×···×I_b, where each I_i is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ?log₂ψ(G)?. In this article, we show that for an interval graph G ?log₂ψ(G)??cub(G)??log₂ψ(G)?+2. It is not clear whether the upper bound of ?log₂ψ(G)?+2 is tight: till now we are unable to find any interval graph with cub(G)>?log₂ψ(G)?. We also show that for an interval graph G, cub(G)??log₂α?, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ?log₂α₂?. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I₁×I₂×···×I_b, where each I_i is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)?cub(G). From the above result, it follows that for any graph G, cub(G)?box(G)?log₂α?. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010     

On the Cubicity of Interval Graphs

A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where R _i (for 1 ≤ i ≤ k) is an interval of the form [a _i , a _i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to .     

On the diameter of i‐center in a graph without long induced paths

A graph G is said to be P_t‐free if it does not contain an induced path on t vertices. The i‐center C_i(G) of a connected graph G is the set of vertices whose distance from any vertex in G is at most i. Denote by I(t) the set of natural numbers i, ⌊t/2⌋ ≤ i ≤ t − 2, with the property that, in every connected P_t‐free graph G, the i‐center C_i(G) of G induces a connected subgraph of G. In this article, the sharp upper bound on the diameter of G[C_i(G)] is established for every i ∈ I(t). The sharp lower bound on I(t) is obtained consequently. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 235–241, 1999     

P3-isomorphisms for graphs

The P₃-graph of a finite simple graph G is the graph whose vertices are the 3-vertex paths of G, with adjacency between two such paths whenever their union is a 4-vertex path or a 3-cycle. In this paper we show that connected fnite simple graphs G and H with isomorphic P₃-graphs are either isomorphic or part of three exceptional families. We also characterize all isomorphisms between P₃-graphs in terms of the original graphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26:35–51, 1997     

Path chromatic numbers of graphs

Let the finite, simple, undirected graph G = (V(G), E(G)) be vertex-colored. Denote the distinct colors by 1,2,…,c. Let V_i be the set of all vertices colored j and let <V_i be the subgraph of G induced by V_i. The k-path chromatic number of G, denoted by χ(G; P_k), is the least number c of distinct colors with which V(G) can be colored such that each connected component of V_i is a path of order at most k, 1 ? i ? c. We obtain upper bounds for χ(G; P_k) and χ(G; P_∞) for regular, planar, and outerplanar graphs.     

A note on on-line ranking number of graphs

A k-ranking of a graph G = (V, E) is a mapping ϕ: V → {1, 2, ..., k} such that each path with end vertices of the same colour c contains an internal vertex with colour greater than c. The ranking number of a graph G is the smallest positive integer k admitting a k-ranking of G. In the on-line version of the problem, the vertices v ₁, v ², ..., v _n of G arrive one by one in an arbitrary order, and only the edges of the induced graph G[{v ₁, v ₂, ..., v _i }] are known when the colour for the vertex v _i has to be chosen. The on-line ranking number of a graph G is the smallest positive integer k such that there exists an algorithm that produces a k-ranking of G for an arbitrary input sequence of its vertices. We show that there are graphs with arbitrarily large difference and arbitrarily large ratio between the ranking number and the on-line ranking number. We also determine the on-line ranking number of complete n-partite graphs. The question of additivity and heredity is discussed as well.     

Threshold characterization of graphs with dilworth number two

A graph with nodes 1, …, n is a threshold signed graph if one can find two positive real numbers S, T and real numbers a₁, …, a_n associated with the vertices in such a way that i, j are linked iff either |a_i + a_j| ≥ S or |a_i - a_j| ≥ T. Such graphs generalize threshold graphs. It is shown that these graphs are precisely the graphs with Dilworth number at most two (the Dilworth number is the maximum number of pairwise incomparable vertices in the vicinal preorder). Some other properties of this subclass of perfect graphs are also presented. The graphs considered in this paper are finite simple graphs G = (V, E), where V is the vertex set of G and E a subset of pairs of G. For x V, N(x) denotes the neighbor set of x: N(x) = {y | {x, y} E}.     

SIR epidemics on random graphs with a fixed degree sequence

Let Δ > 1 be a fixed positive integer. For \begin{align*}{\textbf{ {z}}} \in \mathbb{R}_+^\Delta\end{align*} let G_z be chosen uniformly at random from the collection of graphs on ∥z∥₁n vertices that have z_in vertices of degree i for i = 1,…,Δ. We determine the likely evolution in continuous time of the SIR model for the spread of an infectious disease on G_z, starting from a single infected node. Either the disease halts after infecting only a small number of nodes, or an epidemic spreads to infect a linear number of nodes. Conditioning on the event that more than a small number of nodes are infected, the epidemic is likely to follow a trajectory given by the solution of an associated system of ordinary differential equations. These results also give the likely number of nodes infected during the course of the epidemic and the likely length in time of the epidemic. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Colorful monochromatic connectivity

An edge-coloring of a connected graph is monochromatically-connecting if there is a monochromatic path joining any two vertices. How “colorful” can a monochromatically-connecting coloring be? Let mc(G) denote the maximum number of colors used in a monochromatically-connecting coloring of a graph G. We prove some nontrivial upper and lower bounds for mc(G) and relate it to other graph parameters such as the chromatic number, the connectivity, the maximum degree, and the diameter.     

A NOTE ON GRAPHS WHOSE DESTRUCTIONS YIELD COMPLETE SUBGRAPHS

Abstract

A connected graph G of order p =|V| and sise q =| E | is said to be (a_i, b_i)-destructible (with respect to E_i and V_i say) if a_i,b_i are integral factors of p and an a_i-set of edges E_i exists whose removal from G results in exactly b_i components isomorphic to K_i i.e. whose removal from G isolates the vertices in a b_i-set V_i. The operation of removing E_i and V_i from G results in either Ø or a subgraph H of G and is called an (a_i , b_i)-destruction of G. In this paper we show that the only graphs whose every (a_i,b_i)- destruction results in a complete subgraph are K (1,2) and K₄—e, where e ε K₄.     

Can visibility graphs Be represented compactly?

We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graphG, a familyG={G ₁,G ₂,...,G _k } is called aclique cover ofG if (i) eachG _i is a clique or a bipartite clique, and (ii) the union ofG _i isG. The size of the clique coverG is defined as ∑ _i=1 ^k n _i , wheren _i is the number of vertices inG _i . Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n ²/log² n). An upper bound ofO(n ²/logn) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of sizeO(nlog³ n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn). The work by the first author was supported by National Science Foundation Grant CCR-91-06514. The work by the second author was supported by a USA-Israeli BSF grant. The work by the third author was supported by National Science Foundation Grant CCR-92-11541.