Circular game chromatic number of graphs

In a circular r-colouring game on G, Alice and Bob take turns colouring the vertices of G with colours from the circle S(r) of perimeter r. Colours assigned to adjacent vertices need to have distance at least 1 in S(r). Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χ_cg(G) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r-colouring game on G. This paper proves that for any graph G, , where is the game colouring number of G. This upper bound is shown to be sharp for forests. It is also shown that for any graph G, χ_cg(G)≤2χ_a(G)(χ_a(G)+1), where χ_a(G) is the acyclic chromatic number of G. We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles.     

The total chromatic number of Pseudo-Halin graphs with lower degree

The total chromatic number χ_T(G) of a graph G is the least number of colors needed to color the vertices and the edges of G such that no adjacent or incident elements receive the same color. The Total Coloring Conjecture(TCC) states that for every simple graph G, χ_T(G)≤Δ(G)+2. In this paper, we show that χ_T(G)=Δ(G)+1 for all pseudo-Halin graphs with Δ(G)=4 and 5.     

(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.     

Approximation algorithms for the minimum rainbow subgraph problem

We consider the minimum rainbow subgraph problem (MRS): given a graph G, whose edges are coloured with p colours. Find a subgraph F⊆G of G of minimum order and with p edges such that each colour occurs exactly once. For graphs with maximum degree Δ(G) there is a greedy polynomial-time approximation algorithm for the MRS problem with an approximation ratio of Δ(G). In this paper we present a polynomial-time approximation algorithm with an approximation ratio of for Δ≥2.     

Edge colouring by total labellings

We introduce the concept of an edge-colouring total k-labelling. This is a labelling of the vertices and the edges of a graph G with labels 1,2,…,k such that the weights of the edges define a proper edge colouring of G. Here the weight of an edge is the sum of its label and the labels of its two endvertices. We define to be the smallest integer k for which G has an edge-colouring total k-labelling. This parameter has natural upper and lower bounds in terms of the maximum degree Δ of . We improve the upper bound by 1 for every graph and prove . Moreover, we investigate some special classes of graphs.     

General neighbour-distinguishing index via chromatic number

An edge colouring of a graph G without isolated edges is neighbour-distinguishing if any two adjacent vertices have distinct sets consisting of colours of their incident edges. The general neighbour-distinguishing index of G is the minimum number of colours in a neighbour-distinguishing edge colouring of G. Gy?ri et al. [E. Gy?ri, M. Horňák, C. Palmer, M. Wo?niak, General neighbour-distinguishing index of a graph, Discrete Math. 308 (2008) 827-831] proved that provided G is bipartite and gave a complete characterisation of bipartite graphs according to their general neighbour-distinguishing index. The aim of this paper is to prove that if χ(G)≥3, then . Therefore, if log₂χ(G)∉Z, then .     

Total chromatic number of one kind of join graphs

The total chromatic number χ_T(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if χ_T(G)=Δ(G)+1. In this paper we prove that the join of a complete inequibipartite graph K_n1,n2 and a path P_m is of Type 1.     

Colouring games on outerplanar graphs and trees

Let f be a graph function which assigns to each graph H a non-negative integer f(H)≤|V(H)|. The f-game chromatic number of a graph G is defined through a two-person game. Let X be a set of colours. Two players, Alice and Bob, take turns colouring the vertices of G with colours from X. A partial colouring c of G is legal (with respect to graph function f) if for any subgraph H of G, the sum of the number of colours used in H and the number of uncoloured vertices of H is at least f(H). Both Alice and Bob must colour legally (i.e., the partial colouring produced needs to be legal). The game ends if either all the vertices are coloured or there are uncoloured vertices with no legal colour. In the former case, Alice wins the game. In the latter case, Bob wins the game. The f-game chromatic number of G, χ_g(f,G), is the least number of colours that the colour set X needs to contain so that Alice has a winning strategy. Let be the graph function defined as , for any n≥3 and otherwise. Then is called the acyclic game chromatic number of G. In this paper, we prove that any outerplanar graph G has acyclic game chromatic number at most 7. For any integer k, let ?_k be the graph function defined as ?_k(K₂)=2 and ?_k(P_k)=3 (P_k is the path on k vertices) and ?_k(H)=0 otherwise. This paper proves that if k≥8 then for any tree T, χ_g(?_k,T)≤9. On the other hand, if k≤6, then for any integer n, there is a tree T such that χ_g(?_k,T)≥n.     

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).     

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 .     

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.     

Chromatic capacity and graph operations

The chromatic capacityχ_cap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f:N→N such that χ_cap(G)?f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show that for any positive integers n and k with k?n/2 there exists a graph G with χ(G)=n and χ_cap(G)=n-k, extending a result of Greene. We obtain bounds on that are tight as r→∞, where is the complete n-partite graph with r vertices in each part. Finally, for any positive integers p and q we construct a graph G with χ_cap(G)+1=χ(G)=p that contains no odd cycles of length less than q.     

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.     

Colouring graphs with bounded generalized colouring number

Given a graph G and a positive integer p, χ_p(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χ_p(G) in terms of the k-colouring number of G for k=2^p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χ_k+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:

(a)

For every positive integer p, χ_p(G) is bounded by a constant for all G∈K.

(b)

For every positive integer k, is bounded by a constant for all G∈K.

It was proved by Nešet?il and Ossona de Mendez that (a) is equivalent to the following:

(c)

For every positive integer q, ∇_q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K.

Therefore (b) and (c) are also equivalent. We shall give a direct proof of this equivalence, by introducing ∇_q−(1/2)(G) and by showing that there is a function F_k such that . This gives an alternate proof of the equivalence of (a) and (c).     

Variable neighborhood search for extremal graphs. 23. On the Randi? index and the chromatic number

Let G=(V,E) be a simple graph with vertex degrees d₁,d₂,…,d_n. The Randi? index R(G) is equal to the sum over all edges (i,j)∈E of weights . We prove several conjectures, obtained by the system AutoGraphiX, relating R(G) and the chromatic number χ(G). The main result is χ(G)≤2R(G). To prove it, we also show that if v∈V is a vertex of minimum degree δ of G, G−v the graph obtained from G by deleting v and all incident edges, and Δ the maximum degree of G, then .     

r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

3-trees with few vertices of degree 3 in circuit graphs

A circuit graph(G,C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F_χ with Euler characteristic χ≥0 has a 3-tree with at most vertices of degree 3, where c_χ is a constant depending only on F_χ.     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let