A Planar linear arboricity conjecture

The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [Math Slovaca 30 (1980), 405–417] stated the Linear Arboricity Conjecture (LAC) that the linear arboricity of any simple graph of maximum degree Δ is either ?Δ/2? or ?(Δ + 1)/2?. In [J. L. Wu, J Graph Theory 31 (1999), 129–134; J. L. Wu and Y. W. Wu, J Graph Theory 58(3) (2008), 210–220], it was proven that LAC holds for all planar graphs. LAC implies that for Δ odd, la(G) = ?Δ/2?. We conjecture that for planar graphs, this equality is true also for any even Δ?6. In this article we show that it is true for any even Δ?10, leaving open only the cases Δ = 6, 8. We present also an O(n logn) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when Δ?9. © 2010 Wiley Periodicals, Inc. J Graph Theory     

Covering a graph with cycles

Let k and n be two integers such that k ≥ 0 and n ≥ 3(k + 1). Let G be a graph of order n with minimum degree at least ?(n + k)/2?. Then G contains k + 1 independent cycles covering all the vertices of G such that k of them are triangles. © 1995, John Wiley & Sons, Inc.     

Bounds for the vertex linear arboricity

The vertex linear arboricity vla(G) of a nonempty graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces a subgraph whose connected components are paths. This paper provides an upper bound for vla(G) of a connected nonempty graph G, namely vla(G) ≦ 1 + ?δ(G)/2? where δ(G) denotes the maximum degree of G. Moreover, if δ(G) is even, then vla(G) = 1 + ?δ(G)/2? if and only if G is either a cycle or a complete graph.     

Pentagons and cycle coverings

Let G be a graph of order n ≥ 5k + 2, where k is a positive integer. Suppose that the minimum degree of G is at least ?(n + k)/2?. We show that G contains k pentagons and a path such that they are vertex‐disjoint and cover all the vertices of G. Moreover, if n ≥ 5k + 7, then G contains k + 1 vertex‐disjoint cycles covering all the vertices of G such that k of them are pentagons. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 194–208, 2007     

Acyclic edge coloring of graphs with maximum degree 4

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 that for any simple and finite graph G, a′(G)?Δ + 2, where Δ=Δ(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Δ(G)?4, with the additional restriction that m?2n?1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m?2n, when Δ(G)?4. It follows that for any graph G if Δ(G)?4, then a′(G)?7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 192–209, 2009     

A new upper bound for the independence number of edge chromatic critical graphs

In 1968, Vizing conjectured that if G is a Δ‐critical graph with n vertices, then α(G)≤n/2, where α(G) is the independence number of G. In this paper, we apply Vizing and Vizing‐like adjacency lemmas to this problem and prove that α(G)<(((5Δ?6)n)/(8Δ?6))<5n/8 if Δ≥6. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 202‐212, 2011     

The number of spanning trees in regular graphs

Let C(G) denote the number of spanning trees of a graph G. It is shown that there is a function ?(k) that tends to zero as k tends to infinity such that for every connected, k-regular simple graph G on n vertices C(G) = {k[1 ? δ(G)]}ⁿ. where 0 ≤ δ(G) ≤ ?(k).     

The chromatic index of graphs with a spanning star

Vizing's Theorem states that any graph G has chromatic index either the maximum degree Δ(G) or Δ(G) + 1. If G has 2_s + 1 points and Δ(G) = 2s, a well-known necessary condition for the chromatic index to equal 2_s is that G have at most 2s² lines. Hilton conjectured that this condition is also sufficient. We present a proof of that conjecture and a corollary that helps determine the chromatic index of some graphs with 2s points and maximum degree 2s ? 2.     

The connectivity of large digraphs and graphs

This paper studies the relation between the connectivity and other parameters of a digraph (or graph), namely its order n, minimum degree δ, maximum degree Δ, diameter D, and a new parameter l_pi;, 0 ≤ π ≤ δ ? 2, related with the number of short paths (in the case of graphs l₀ = ?(g ? 1)/2? where g stands for the girth). For instance, let G = (V,A) be a digraph on n vertices with maximum degree Δ and diameter D, so that n ≤ n(Δ, D) = 1 + Δ + Δ ² + … + Δ^D (Moore bound). As the main results it is shown that, if κ and λ denote respectively the connectivity and arc-connectivity of G, . Analogous results hold for graphs. © 1993 John Wiley & Sons, Inc.     

An upper bound for the total chromatic number of dense graphs

In this paper we consider those graphs that have maximum degree at least 1/k times their order, where k is a (small) positive integer. A result of Hajnal and Szemerédi concerning equitable vertex-colorings and an adaptation of the standard proof of Vizing's Theorem are used to show that if the maximum degree of a graph G satisfies Δ(G) ≥ |V(G)/k, then X″(G) ≤ Δ(G) + 2k + 1. This upper bound is an improvement on the currently available upper bounds for dense graphs having large order.     

Entire colouring of plane graphs

It was conjectured by Kronk and Mitchem in 1973 that simple plane graphs of maximum degree Δ are entirely (Δ+4)-colourable, i.e., the vertices, edges, and faces of a simple plane graph may be simultaneously coloured with Δ+4 colours in such a way that adjacent or incident elements are coloured by distinct colours. Before this paper, the conjecture has been confirmed for Δ?3 and Δ?6 (the proof for the Δ=6 case has a correctable error). In this paper, we settle the whole conjecture in the positive. We prove that if G is a plane graph with maximum degree 4 (parallel edges allowed), then G is entirely 8-colourable. If G is a plane graph with maximum degree 5 (parallel edges allowed), then G is entirely 9-colourable.     

Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

The Augmentation Quotients of the Groups of Order 25, I

In this article we present the nth power Δ ⁿ (G) of the augmentation ideal Δ(G) and describe the structure of Q _n (G) = Δ ⁿ (G)/Δ ⁿ⁺¹(G) for 35 particular groups G of order 2⁵. The structure of Q _n (G) for all the remaining groups of order 2⁵ will be determined in a forthcoming article.     

A new upper bound on the cyclic chromatic number

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χ^c. Let Δ^* be the maximum face degree of a graph. There exist plane graphs with χ^c = ?3/2 Δ^*?. Ore and Plummer [ 5 ] proved that χ^c ≤ 2, Δ^*, which bound was improved to ?9/5, Δ^*? by Borodin, Sanders, and Zhao [ 1 ], and to ?5/3,Δ^*? by Sanders and Zhao [ 7 ]. We introduce a new parameter k^*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χ^c ≤ max {Δ^* + 3,k^* + 2, Δ^* + 14, 3, k^* + 6, 18}, and if Δ^* ≥ 4 and k^* ≥ 4, then χ^c ≤ Δ^* + 3,k^* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Graphic Sequences Have Realizations Containing Bisections of Large Degree

A bisection of a graph is a balanced bipartite spanning sub‐graph. Bollobás and Scott conjectured that every graph G has a bisection H such that deg_H(v) ≥ ?deg_G(v)/2? for all vertices v. We prove a degree sequence version of this conjecture: given a graphic sequence π, we show that π has a realization G containing a bisection H where deg_H(v) ≥ ?(deg_G(v) ? 1)/2? for all vertices v. This bound is very close to best possible. We use this result to provide evidence for a conjecture of Brualdi (Colloq. Int. CNRS, vol. 260, CNRS, Paris) and Busch et al. (2011), that if π and π ? k are graphic sequences, then π has a realization containing k edge‐disjoint 1‐factors. We show that if the minimum entry δ in π is at least n/2 + 2, then π has a realization containing edge‐disjoint 1‐factors. We also give a construction showing the limits of our approach in proving this conjecture. © 2011 Wiley Periodicals, Inc. J Graph Theory     

N-extendability of symmetric graphs

It is proved that a cyclically (k ? 1)(2n ? 1)-edge-connected edge transitive k-regular graph with even order is n-extendable, where k ≥ 3 and k ? 1 ≥ n ≥ ?(k + 1)/2?. The bound of cyclic edge connectivity is sharp when k = 3. © 1993 John Wiley & Sons, Inc.     

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     

Cycle and cocycle coverings of graphs

In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family ?? of at most n?1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and ε edges having cogirth g^*?3 and k(G) components, there is a family of at most ε?n+k(G) cocycles which cover the edges of G at least twice. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 270–284, 2010     

Inequalities for the first‐fit chromatic number

The First‐Fit (or Grundy) chromatic number of G, written as χ_FF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well‐known Nordhaus‐‐Gaddum inequality states that the sum of the ordinary chromatic numbers of an n‐vertex graph and its complement is at most n + 1. Zaker suggested finding the analogous inequality for the First‐Fit chromatic number. We show for n ≥ 10 that ?(5n + 2)/4? is an upper bound, and this is sharp. We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases. We also show that the smallest order of C₄‐free bipartite graphs with χ_FF(G) = k is asymptotically 2k² (the upper bound answers a problem of Zaker [9]). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 75–88, 2008