On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

Improvement on Brooks' chromatic bound for a class of graphs

Brooks' Theorem says that if for a graph G,Δ(G)=n, then G is n-colourable, unless (1) n=2 and G has an odd cycle as a component, or (2) n>2 and K_n+1 is a component of G. In this paper we prove that if a graph G has none of some three graphs (K_1,3;K₅?e and H) as an induced subgraph and if Δ(G)?6 and d(G)<Δ(G), then χ(G)<Δ(G). Also we give examples to show that the hypothesis Δ(G)?6 can not be non-trivially relaxed and the graph K₅?e can not be removed from the hypothesis. Moreover, for a graph G with none of K_1,3;K₅?e and H as an induced subgraph, we verify Borodin and Kostochka's conjecture that if for a graph G,Δ(G)?9 and d(G)<Δ(G), then χ(G)<Δ(G).     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Weak sense of direction labelings and graph embeddings

An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences α and α^′ generated by non-trivial walks on G starting at u, f(α)=f(α^′) if and only if the two walks end at the same node. The function f is referred to as a coding function of λ. The weak sense of direction number of G, WSD(G), is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD(G)≥Δ⁺(G), where Δ⁺(G) is the maximum outdegree of G.Let us say that a function τ:V(G)→V(H) is an embedding from G onto H if τ demonstrates that G is isomorphic to a subgraph of H. We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f_H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f_H as a coding function if and only if G can be embedded onto H. Additionally, we show that the problem “Given G, does G have a WSD-labeling that uses a particular coding function f?” is NP-complete even when G and f are fairly simple.Second, when D is a distributive lattice, H(D) is its Hasse diagram and G(D) is its cover graph, then WSD(H(D))=Δ⁺(H(D))=d^∗, where d^∗ is the smallest integer d so that H(D) can be embedded onto the d-dimensional mesh. Along the way, we also prove that the isometric dimension of G(D) is its diameter, and the lattice dimension of G(D) is Δ⁺(H(D)). Our WSD-labelings are poset-based, making use of Birkhoff’s characterization of distributive lattices and Dilworth’s theorem for posets.     

Edge-colorings avoiding rainbow and monochromatic subgraphs

For two graphs G and H, let the mixed anti-Ramsey numbers, maxR(n;G,H), (minR(n;G,H)) be the maximum (minimum) number of colors used in an edge-coloring of a complete graph with n vertices having no monochromatic subgraph isomorphic to G and no totally multicolored (rainbow) subgraph isomorphic to H. These two numbers generalize the classical anti-Ramsey and Ramsey numbers, respectively.We show that maxR(n;G,H), in most cases, can be expressed in terms of vertex arboricity of H and it does not depend on the graph G. In particular, we determine maxR(n;G,H) asymptotically for all graphs G and H, where G is not a star and H has vertex arboricity at least 3.In studying minR(n;G,H) we primarily concentrate on the case when G=H=K₃. We find minR(n;K₃,K₃) exactly, as well as all extremal colorings. Among others, by investigating minR(n;K_t,K₃), we show that if an edge-coloring of K_n in k colors has no monochromatic K_t and no rainbow triangle, then n?2^kt2.     

A survey on labeling graphs with a condition at distance two

For positive integers k,d₁,d₂, a k-L(d₁,d₂)-labeling of a graph G is a function f:V(G)→{0,1,2,…,k} such that |f(u)-f(v)|?d_i whenever the distance between u and v is i in G, for i=1,2. The L(d₁,d₂)-number of G, λ_d1,d2(G), is the smallest k such that there exists a k-L(d₁,d₂)-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.     

On minimally circular-imperfect graphs

A circular-perfect graph is a graph of which each induced subgraph has the same circular chromatic number as its circular clique number. In this paper, (1) we prove a lower bound on the order of minimally circular-imperfect graphs, and characterize those that attain the bound; (2) we prove that if G is a claw-free minimally circular-imperfect graph such that ω_c(G-x)>ω(G-x) for some x∈V(G), then G=K_(2k+1)/2+x for an integer k; and (3) we also characterize all minimally circular-imperfect line graphs.     

A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ?λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ^*. Define λ_c=1/(k?1)×(k/(k?1)) ^k . Then: (i) for λ≤λ_c, the Gibbs measure is unique (and coincides with the above measure μ^*), (ii) for λ>λ_c, in addition to μ^*, there exist two distinct translation-periodic measures, μ₊and μ_?, taken to each other by the unit space shift. Measures μ₊and μ_?are extreme ?λ>λ_c. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $ , measure μ^*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λ_eand λ_o, for even and odd sites. We discuss open problems and state several related conjectures.     

Optimal graphs for chromatic polynomials

Let F(n,e) be the collection of all simple graphs with n vertices and e edges, and for G∈F(n,e) let P(G;λ) be the chromatic polynomial of G. A graph G∈F(n,e) is said to be optimal if another graph H∈F(n,e) does not exist with P(H;λ)?P(G;λ) for all λ, with strict inequality holding for some λ. In this paper we derive necessary conditions for bipartite graphs to be optimal, and show that, contrarily to the case of lower bounds, one can find values of n and e for which optimal graphs are not unique. We also derive necessary conditions for bipartite graphs to have the greatest number of cycles of length 4.     

A note on cycle spectra of line graphs

We show that line graphs G=L(H) with σ₂(G)≥7 contain cycles of all lengths k, 2rad(H)+1≤k≤c(G). This implies that every line graph of such a graph with 2rad(H)≥Δ(H) is subpancyclic, improving a recent result of Xiong and Li. The bound on σ₂(G) is best possible.     

Characterization of Gromov hyperbolic short graphs

To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs(in an informal way,a graph G is r-short if the shortcuts in the cycles of G have length less than r):an r-short graph G is hyperbolic if and only if S9r(G)is finite,where SR(G):=sup{L(C):C is an R-isometric cycle in G}and we say that a cycle C is R-isometric if dC(x,y)≤dG(x,y)+R for every x,y∈C.     

Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

Labelling of some planar graphs with a condition at distance two

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale’s channel assignment problem, which was first explored by Griggs and Yeh. For positive integerp ≥q, the λ _p,q -number of graph G, denoted λ(G;p, q), is the smallest span among all integer labellings ofV(G) such that vertices at distance two receive labels which differ by at leastq and adjacent vertices receive labels which differ by at leastp. Van den Heuvel and McGuinness have proved that λ(G;p, q) ≤ (4q-2) Δ+10p+38q-24 for any planar graphG with maximum degree Δ. In this paper, we studied the upper bound of λ _p ,q-number of some planar graphs. It is proved that λ(G;p, q) ≤ (2q?1)Δ + 2(2p?1) ifG is an outerplanar graph and λ(G;p,q) ≤ (2q?1) Δ + 6p - 4q - 1 if G is a Halin graph.     

The strong chromatic index of a class of graphs

The strong chromatic index of a graph G is the minimum integer k such that the edge set of G can be partitioned into k induced matchings. Faudree et al. [R.J. Faudree, R.H. Schelp, A. Gyárfás, Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205-211] proposed an open problem: If G is bipartite and if for each edge xy∈E(G), d(x)+d(y)≤5, then sχ^′(G)≤6. Let H₀ be the graph obtained from a 5-cycle by adding a new vertex and joining it to two nonadjacent vertices of the 5-cycle. In this paper, we show that if G (not necessarily bipartite) is not isomorphic to H₀ and d(x)+d(y)≤5 for any edge xy of G then sχ^′(G)≤6. The proof of the result implies a linear time algorithm to produce a strong edge coloring using at most 6 colors for such graphs.     

Competitively Tight Graphs

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph G is related to the edge clique cover number θ _E (G) of the graph G via θ _E (G) ? |V(G)| + 2 ≤ k(G) ≤ θ _E (G). We first show that for any positive integer m satisfying 2 ≤ m ≤ |V(G)|, there exists a graph G with k(G) = θ _E (G) ? |V(G)| + m and characterize a graph G satisfying k(G) = θ _E (G). We then focus on what we call competitively tight graphs G which satisfy the lower bound, i.e., k(G) = θ _E (G) ? |V(G)| + 2. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.     

1‐Factorizations of pseudorandom graphs

A 1‐factorization of a graph G is a collection of edge‐disjoint perfect matchings whose union is E(G). In this paper, we prove that for any ?>0, an (n,d,λ)‐graph G admits a 1‐factorization provided that n is even, C₀ ≤ d ≤ n?1 (where C₀=C₀(?) is a constant depending only on ?), and λ ≤ d^1??. In particular, since (as is well known) a typical random d‐regular graph G_n,d is such a graph, we obtain the existence of a 1‐factorization in a typical G_n,d for all C₀ ≤ d ≤ n?1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1‐factorizations of such graphs G, which is better by a factor of 2^nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.