Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

Lower bounds of the Laplacian graph eigenvalues

In this paper we prove that all positive eigenvalues of the Laplacian of an arbitrary simple graph have some positive lower bounds. For a fixed integer k 1 we call a graph without isolated vertices k-minimal if its kth greatest Laplacian eigenvalue reaches this lower bound. We describe all 1-minimal and 2-minimal graphs and we prove that for every k 3 the path P_k+1 on k + 1 vertices is the unique k-minimal graph.     

The diameter of domination k-critical graphs

A graph is k-domination-critical if γ(G) = k, and for any edge e not in G, γ(G + e) = k ? 1. In this paper we show that the diameter of a domination k-critical graph with k ≧ 2 is at most 2k ? 2. We also show that for every k ≧ 2, there is a k-domination-critical graph having diameter [(3/2)k ? 1]. We also show that the diameter of a 4-domination-critical graph is at most 5.     

Maximally connected digraphs

This paper introduces a new parameter I = I(G) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let k, λ, δ, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that λ = δ if D ? 2I and κ k = δ if D ? 2I - 1. Analogous results involving upper bounds for k and λ are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-λ and super-k are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized.     

Set intersection representations for almost all graphs

Two variations of set intersection representation are investigated and upper and lower bounds on the minimum number of labels with which a graph may be represented are found that hold for almost all graphs. Specifically, if θ_k(G) is defined to be the minimum number of labels with which G may be represented using the rule that two vertices are adjacent if and only if they share at least k labels, there exist positive constants c_k and c′_k such that almost every graph G on n vertices satisfies Changing the representation only slightly by defining θ;_odd (G) to be the minimum number of labels with which G can be represented using the rule that two vertices are adjacent if and only if they share an odd number of labels results in quite different behavior. Namely, almost every graph G satisfies Furthermore, the upper bound on θ_odd(G) holds for every graph. © 1996 John Wiley & Sons, Inc.     

Maximal triangle-free graphs with restrictions on the degrees

We investigate the problem that at least how many edges must a maximal triangle-free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n ? 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence c_k → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cn^?, 1/2 < ? < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n - 1)^1/2 is impossible in a maximal triangle-free graph.)     

Pancyclic and panconnected line graphs

Let G be a graph of order n ? 3. We prove that if G is k-connected (k ? 2) and the degree sum of k + 1 mutually independent vertices of G is greater than 1/3(k + 1)(n + 1), then the line graph L(G) of G is pancyclic. We also prove that if G is such that the degree sum of every 2 adjacent vertices is at least n, then L(G) is panconnected with some exceptions.     

Acyclic edge coloring of 2‐degenerate graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2‐degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2‐degenerate graphs properly contains series–parallel graphs, outerplanar graphs, non ? regular subcubic graphs, planar graphs of girth at least 6 and circle graphs of girth at least 5 as subclasses. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G)?Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. We prove the conjecture for 2‐degenerate graphs. In fact we prove a stronger bound: we prove that if G is a 2‐degenerate graph with maximum degree Δ, then a′(G)?Δ + 1. © 2010 Wiley Periodicals, Inc. J Graph Theory 69: 1–27, 2012     

Large P4-free graphs with bounded degree

Let ex * (D; H) denote the maximum number of edges in a connected graph with maximum degree D and no induced subgraph isomorphic to H. We prove that this is finite only when H is a disjoint union of paths,m in which case we provide crude upper and lower bounds. When H is the four-vertex path P₄, we prove that the complete bipartite graph K_D,D is the unique extremal graph. Furthermore, if G is a connected P₄-free graph with maximum degree D and clique number ω, then G has at most D² ? D(ω ? 2)/2 edges. © 1993 John Wiley & Sons, Inc.     

Boxicity of Circular Arc Graphs

A k-dimensional box is a Cartesian product R ₁ × · · · × R _k where each R _i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN _{3 2}{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.     

Circular choosability of graphs

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) χ_{c, l}(G) of a graph G and prove that they are equivalent. Then we prove that for any graph G, χ_{c, l}(G) ≥ χ_l(G) ? 1. Examples are given to show that this bound is sharp in the sense that for any ? 0, there is a graph G with χ_{c, l}(G) > χ_l(G) ? 1 + ?. It is also proved that k‐degenerate graphs G have χ_{c, l}(G) ≤ 2k. This bound is also sharp: for each ? < 0, there is a k‐degenerate graph G with χ_{c, l}(G) ≥ 2k ? ?. This shows that χ_{c, l}(G) could be arbitrarily larger than χ_l(G). Finally we prove that if G has maximum degree k, then χ_{c, l}(G) ≤ k + 1. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 210–218, 2005     

Maximum matchings in a regular graph of specified connectivity and bounded order

Upper bounds are placed on the order of a k-regular m-connected graph G that produce a lower bound on the number of independent edges in G. As a corollary, we obtain the order of a smallest k-regular m-connected graph which has no 1-factor.     

Some 3‐connected 4‐edge‐critical non‐Hamiltonian graphs

Let γ(G) be the domination number of graph G, thus a graph G is k‐edge‐critical if γ (G) = k, and for every nonadjacent pair of vertices u and υ, γ(G + uυ) = k?1. In Chapter 16 of the book “Domination in Graphs—Advanced Topics,” D. Sumner cites a conjecture of E. Wojcicka under the form “3‐connected 4‐critical graphs are Hamiltonian and perhaps, in general (i.e., for any k ≥ 4), (k?1)‐connected, k‐edge‐critical graphs are Hamiltonian.” In this paper, we prove that the conjecture is not true for k = 4 by constructing a class of 3‐connected 4‐edge‐critical non‐Hamiltonian graphs. © 2005 Wiley Periodicals, Inc.     

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

Posets and planar graphs

A k-decomposition (G₁,…,G_k) of a graph G is a partition of its edge set to form k spanning subgraphs G₁,…,G_k. The classical theorem of Nordhaus and Gaddum bounds χ(G₁) + χ(G₂) and χ(G₁)χ(G₂) over all 2-decompositions of K_n. For a graph parameter p, let p(k;G) denote the maximum of over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list chromatic number χℓ, and Szekeres–Wilf number σ satisfy ω(2;K_n) = χ(2;K_n) = χℓ(2;K_n) = σ(2;K_n) = n + 1. We obtain lower and upper bounds for ω(k;K_n), χ(k;K_n), χℓ(k;K_n), and σ(k;K_n). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of χ(k;G) over all graphs embedded on a given surface. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Circumference, chromatic number and online coloring

Erd?s conjectured that if G is a triangle free graph of chromatic number at least k≥3, then it contains an odd cycle of length at least k ^2?o(1) [13,15]. Nothing better than a linear bound ([3], Problem 5.1.55 in [16]) was so far known. We make progress on this conjecture by showing that G contains an odd cycle of length at least Ω(k log logk). Erd?s’ conjecture is known to hold for graphs with girth at least five. We show that if a graph with girth four is C ₅ free, then Erd?s’ conjecture holds. When the number of vertices is not too large we can prove better bounds on χ. We also give bounds on the chromatic number of graphs with at most r cycles of length 1 mod k, or at most s cycles of length 2 mod k, or no cycles of length 3 mod k. Our techniques essentially consist of using a depth first search tree to decompose the graph into ordered paths, which are then fed to an online coloring algorithm. Using this technique we give simple proofs of some old results, and also obtain several other results. We also obtain a lower bound on the number of colors which an online coloring algorithm needs to use to color triangle free graphs.     

Covering the vertices of a graph by cycles of prescribed length

The main theorem of that paper is the following: let G be a graph of order n, of size at least (n² - 3n + 6)/2. For any integers k, n₁, n₂,…,n_k such that n = n₁ + n₂ +. + n_k and n_i ? 3, there exists a covering of the vertices of G by disjoint cycles (C_i) =l…k with |C_i| = ni, except when n = 6, n₁ = 3, n₂ = 3, and G is isomorphic to G₁, the complement of G₁ consisting of a C₃ and a stable set of three vertices, or when n = 9, n₁ = n₂ = n₃ = 3, and G is isomorphic to G₂, the complement of G₂ consisting of a complete graph on four vertices and a stable set of five vertices. We prove an analogous theorem for bipartite graphs: let G be a bipartite balanced graph of order 2n, of size at least n² - n + 2. For any integers s, n₁, n₂,…,n_s with n_i ? 2 and n = n₁ + n₂ + ? + n_s, there exists a covering of the vertices of G by s disjoint cycles C_i, with |C_i| = 2n_i.     

The <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d _k (G). In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d ₁(G).      

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.