Equitable colorings of Kronecker products of graphs

For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f⁻¹(i)|−|f⁻¹(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ₌(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ₌(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles.     

Adaptable choosability of planar graphs with sparse short cycles

Given a (possibly improper) edge colouring F of a graph G, a vertex colouring of G is adapted toF if no colour appears at the same time on an edge and on its two endpoints. A graph G is called (for some positive integer k) if for any list assignment L to the vertices of G, with |L(v)|≥k for all v, and any edge colouring F of G, G admits a colouring c adapted to F where c(v)∈L(v) for all v. This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle.     

Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K₄-minor-free graph with maximum degree Δ, then G is entirely 7-choosable if Δ≤4 and G is entirely (Δ+2)-choosable if Δ≥5; that is, if every vertex, edge and face of G is given a list of max{7,Δ+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K_2,3-minor-free graph or a -minor-free graph. As a special case this proves that the Entire Coluring Conjecture, that a plane graph is entirely (Δ+4)-colourable, holds if G is a plane embedding of a K₄-minor-free graph, a K_2,3-minor-free graph or a -minor-free graph.     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Injective colorings of sparse graphs

Let denote the maximum average degree (over all subgraphs) of G and let χ_i(G) denote the injective chromatic number of G. We prove that if , then χ_i(G)≤Δ(G)+1; and if , then χ_i(G)=Δ(G). Suppose that G is a planar graph with girth g(G) and Δ(G)≥4. We prove that if g(G)≥9, then χ_i(G)≤Δ(G)+1; similarly, if g(G)≥13, then χ_i(G)=Δ(G).     

Total colorings and list total colorings of planar graphs without intersecting 4-cycles

Suppose that G is a planar graph with maximum degree Δ and without intersecting 4-cycles, that is, no two cycles of length 4 have a common vertex. Let χ^″(G), and denote the total chromatic number, list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that χ^″(G)=Δ+1 if Δ≥7, and and if Δ(G)≥8. Furthermore, if G is a graph embedded in a surface of nonnegative characteristic, then our results also hold.     

Linkability in iterated line graphs

We prove that for every graph H with the minimum degree δ?5, the third iterated line graph L³(H) of H contains as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number k_G such that for every i?k_G the i-iterated line graph of G is -linked. Since the degree of Lⁱ(G) is even, the result is best possible.     

Hamilton-connected indices of graphs

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131-148] defined hc(G) to be the least integer m such that the iterated line graph L^m(G) is Hamilton-connected. Let be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degree 2 in G. In this paper, we show that . We also show that κ³(G)≤hc(G)≤κ³(G)+2 where κ³(G) is the least integer m such that L^m(G) is 3-connected. Finally we prove that hc(G)≤|V(G)|−Δ(G)+1. These bounds are all sharp.     

Precoloring extension for 2-connected graphs with maximum degree three

Let G=G(V,E) be a simple graph, L a list assignment with |L(v)|=Δ(G)∀v∈V and W⊆V an independent subset of the vertex set. Define to be the minimum distance between two vertices of W. In this paper it is shown that if G is 2-connected with Δ(G)=3 and G is not K₄ then every precoloring of W is extendable to a proper list coloring of G provided that d(W)≥6. An example shows that the bound is sharp. This result completes the investigation of precoloring extensions for graphs with |L(v)|=Δ(G) for all v∈V where the precolored set W is an independent set.     

Acyclic edge colouring of planar graphs without short cycles

Let G=(V,E) be any finite graph. A mapping C:E→[k] is called an acyclic edgek-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i or j, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by .In 2001, Alon et al. conjectured that for any graph G it holds that ; here Δ(G) stands for the maximum degree of G.In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4,6,8 and 9. We also show that if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then .     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

(2,1)-Total labelling of outerplanar graphs

The (2,1)-total labelling number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. In this paper we prove that if G is an outerplanar graph with maximum degree Δ(G), then if Δ(G)?5, or Δ(G)=3 and G is 2-connected, or Δ(G)=4 and G contains no intersecting triangles.     

A result on the total colouring of powers of cycles

The total chromatic number χ_T(G) is the least number of colours needed to colour the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same colour. The Total Colouring Conjecture (TCC) states that for every simple graph G, χ_T(G)?Δ(G)+2. This work verifies the TCC for powers of cycles even and 2<k<n/2, showing that there exists and can be polynomially constructed a (Δ(G)+2)-total colouring for these graphs.     

Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

Self-correspondences of the generic fibre of Lefschetz pencils and the Leray filtration

In this paper we give a detailed analysis of the interaction between homological self-correspondences of the general fibre Y/k(t) of the Lefschetz fibration of a Lefschetz pencil on a smooth projective variety X/k, and the Leray filtration of ρ. We derive the result that, if the standard conjecture B(Y) holds, then the operator is algebraic, where is defined as the inverse of L on LPn−1(X) and 0 on LkPj(X) for (1,n−1)≠(k,j); in the course of our proof we see that, under the above assumption, the Künneth projectors for i≠n−1,n,n+1 are algebraic.     

Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R₁×R₂×?×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 is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G)≤t+⌈log(n−t)⌉−1 and , where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds.F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, and , where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then and this bound is tight. We also show that if G is a bipartite graph then . We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to . Interestingly, if boxicity is very close to , then chromatic number also has to be very high. In particular, we show that if , s≥0, then , where χ(G) is the chromatic number of G.