The splitting number of the complete graph in the projective plane

If a given graphG can be obtained bys vertex identifications from a suitable graph embeddable in the projective plane ands is the minimum number for which this is possible thens is called the splitting number ofG in the projective plane. Here a formula for the splitting number of the complete graph in the projective plane is derived.     

Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.     

Retarded regular graphs are regular or semiregular

A graph G is said to be retarded regular if there is a positive integral number s such that the number of walks of length s starting at vertices of G is a constant function. Regular and semiregular graphs are retarded regular with s?=?1 and s\!≤ \!2, respectively. We prove that any retarded regular connected graph is either regular or semiregular.     

Lower bounds for lower Ramsey numbers

For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as well as determine precisely the values s(1, n).     

The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

Star coloring of sparse graphs

A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induces a star forest. The star chromatic number χ_s(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χ_s(G) and the maximum average degree Mad(G) of a graph G. We prove that:

• 1. If G is a graph with , then χ_s(G)≤4.
• 2. If G is a graph with and girth at least 6, then χ_s(G)≤5.
• 3. If G is a graph with and girth at least 6, then χ_s(G)≤6.

These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 201–219, 2009     

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.     

Steiner Numbers in Graphs

Abstract

The Steiner distance d(S) of a set S of vertices in a connected graph G is the minimum size of a connected subgraph of G that contains S. The Steiner number s(G) of a connected graph G of order p is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = p—1. A smallest set S of vertices of a connected graph G of order p for which d(S) = p—1 is called a Steiner spanning set of G. It is shown that every connected graph has a unique Steiner spanning set. If G is a connected graph of order p and k is an integer with 0 ≤ k ≤ p—1, then the kth Steiner number sk(G) of G is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = k. The sequence s_o(G),s₁ (G),…,8_p-1(G) is called the Steiner sequence of G. Steiner sequences for trees are characterized.     

The minimum degree of Ramsey‐minimal graphs

We write H → G if every 2‐coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G‐minimal if H → G, but for every proper subgraph H′ of H, H′ ? G. We define s(G) to be the minimum s such that there exists a G‐minimal graph with a vertex of degree s. We prove that s(K_k) = (k ? 1)² and s(K_a,b) = 2 min(a,b) ? 1. We also pose several related open problems. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 167–177, 2007     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

An <Emphasis Type="Italic">s</Emphasis>-Hamiltonian Line Graph Problem

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.     

On minimum degree in Hamiltonian path graphs

For integers p and s satisfying 2 ? s ? p ? 1, let m(p,s) denote the maximum number of edges in a graph G of order p such that the minimum degree in the hamiltonian path graph of G equals s. The values of m(p, s) are determined for 2 ? s ? p/2 and for (2p ? 2)/3 ? s ? p ? 1, and upper and lower bounds on m(p, s) are obtained for p/2 < s < (2p ? 2)/3.     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

Irredundant ramsey numbers for graphs

The irredundant Ramsey number s(m, n) is the least value of p such that for any p-vertex graph G, either G has an irredundant set of at least n vertices or its complement G has an irredundant set of at least m vertices. The existence of these numbers is guaranteed by Ramsey's theorem. We prove that s(3, 3) = 6, s(3, 4) = 8, and s(3,5) = 12.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

On s‐Hamiltonian Line Graphs

For an integer s ≥ 0, a graph G is s‐hamiltonian if for any vertex subset with |S^′| ≤ s, G ‐ S^′ is hamiltonian. It is well known that if a graph G is s‐hamiltonian, then G must be (s+2)‐connected. The converse is not true, as there exist arbitrarily highly connected nonhamiltonian graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s‐hamiltonian if and only if it is (s+2)‐connected.     

Powers of edge ideals with linear resolutions

We show that if G is a gap-free and diamond-free graph, then I(G)^s has a linear minimal free resolution for every s≥2.     

The irredundant ramsey number s(3, 7)

The irredundant Ramsey number s(m, n) is the smallest p such that for every graph G with p vertices, either G contains an n-element irredundant set or its complement G contains an m-element irredundant set. Cockayne, Hattingh, and Mynhardt have given a computer-assisted proof that s(3, 7) = 18. The purpose of this paper is to give a self-contained proof of this result. © 1995 John Wiley & Sons, Inc.     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On σ-equivalence and χ-equivalence of graphs

We define a partial ordering on the set of σ-polynomials as well as a vertex splitting operation on the set of graphs, and introduce the notions of σ-equivalence and σ-uniqueness of graphs. Let σ(G) be the σ-polynomial of a graph G and (OVERBAR)σ(G) = σ(G^c). Let H = (G, v, A, B) be a vertex splitting graph of G. We prove that (OVERBAR)σ(G) ≤ (OVERBAR)σ(H) and the equality holds if and only if every vertex of A is adjacent to every vertex of B. This gives us an effective means to find σ-equivalent and χ-equivalent graphs. A necessary and sufficient condition for a graph to be χ-unique but not σ-unique is also obtained. © 1996 John Wiley & Sons, Inc.