On the oriented chromatic number of graphs with given excess

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a directed cycle on 5 vertices which have oriented chromatic number 5. This bound is tight for k?4.     

Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs

A graph is planar if it can be embedded on the plane without edge-crossings. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239–249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215–219, 2007].     

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 .     

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k∈N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree Δ where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most Δ+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ+1. We also present examples where these bounds are attained.Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most .     

Linear coloring of planar graphs with large girth

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of the graph G is the smallest number of colors in a linear coloring of G. In this paper we prove that every planar graph G with girth g and maximum degree Δ has if G satisfies one of the following four conditions: (1) g≥13 and Δ≥3; (2) g≥11 and Δ≥5; (3) g≥9 and Δ≥7; (4) g≥7 and Δ≥13. Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6.     

The distinguishing chromatic number of Cartesian products of two complete graphs

A labeling of a graph G is distinguishing if it is only preserved by the trivial automorphism of G. The distinguishing chromatic number of G is the smallest integer k such that G has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product is determined for all k and n. In most of the cases it is equal to the chromatic number, thus answering a question of Choi, Hartke and Kaul whether there are some other graphs for which this equality holds.     

Approximating the maximum 3-edge-colorable subgraph problem

We offer the following structural result: every triangle-free graph G of maximum degree 3 has 3 matchings which collectively cover at least of its edges, where γ_o(G) denotes the odd girth of G. In particular, every triangle-free graph G of maximum degree 3 has 3 matchings which cover at least 13/15 of its edges. The Petersen graph, where we can 3-edge-color at most 13 of its 15 edges, shows this to be tight. We can also cover at least 6/7 of the edges of any simple graph of maximum degree 3 by means of 3 matchings; again a tight bound.For a fixed value of a parameter k≥1, the Maximum k-Edge-Colorable Subgraph Problem asks to k-edge-color the most of the edges of a simple graph received in input. The problem is known to be APX-hard for all k≥2. However, approximation algorithms with approximation ratios tending to 1 as k goes to infinity are also known. At present, the best known performance ratios for the cases k=2 and k=3 were 5/6 and 4/5, respectively. Since the proofs of our structural result are algorithmic, we obtain an improved approximation algorithm for the case k=3, achieving approximation ratio of 6/7. Better bounds, and allowing also for parallel edges, are obtained for graphs of higher odd girth (e.g., a bound of 13/15 when the input multigraph is restricted to be triangle-free, and of 19/21 when C₅’s are also banned).     

A note on Thomassen?s conjecture

In 1983 C. Thomassen [8] conjectured that for every k,g∈N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. A result of Pyber, Szemerédi, and the second author implies that the conjecture is true for every graph G with average .We strengthen this and show that the conjecture holds for every graph G with average for some constants α, β depending on k and g.     

The 6-relaxed game chromatic number of outerplanar graphs

Suppose G is a graph and k,d are integers. The (k,d)-relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Alice’s goal is to have all the vertices coloured, and Bob’s goal is the opposite: to have an uncoloured vertex without a legal colour. The d-relaxed game chromatic number of G, denoted by , is the least number k so that when playing the (k,d)-relaxed colouring game on G, Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then for d≥6.     

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.     

Extra edge connectivity and isoperimetric edge connectivity

An edge set S of a connected graph G is a k-extra edge cut, if G-S is no longer connected, and each component of G-S has at least k vertices. The cardinality of a minimum k-extra edge cut, denoted by λ_k(G), is the k-extra edge connectivity of G. The kth isoperimetric edge connectivity γ_k(G) is defined as , where ω(U) is the number of edges with one end in U and the other end in . Write β_k(G)=min{ω(U):U⊂V(G),|U|=k}. A graph G with is said to be γ_k-optimal.In this paper, we first prove that λ_k(G)=γ_k(G) if G is a regular graph with girth g?k/2. Then, we show that except for K_3,3 and K₄, a 3-regular vertex/edge transitive graph is γ_k-optimal if and only if its girth is at least k+2. Finally, we prove that a connected d-regular edge-transitive graph with d?6e_k(G)/k is γ_k-optimal, where e_k(G) is the maximum number of edges in a subgraph of G with order k.     

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.     

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.     

Ohba’s conjecture for graphs with independence number five

Ohba has conjectured that if G is a k-chromatic graph with at most 2k+1 vertices, then the list chromatic number or choosability of G is equal to its chromatic number χ(G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k-partite graph and each part has at most five vertices.     

Longest cycles in k-connected graphs with given independence number

The Chvátal–Erd?s Theorem states that every graph whose connectivity is at least its independence number has a spanning cycle. In 1976, Fouquet and Jolivet conjectured an extension: If G is an n-vertex k-connected graph with independence number a, and a?k, then G has a cycle with length at least . We prove this conjecture.     

Edge covered critical multigraphs

Let G be a multigraph with edge set E(G). An edge coloring C of G is called an edge covered coloring, if each color appears at least once at each vertex v∈V(G). The maximum positive integer k such that G has a k edge covered coloring is called the edge covered chromatic index of G and is denoted by . A graph G is said to be of class if and otherwise of class. A pair of vertices {u,v} is said to be critical if . A graph G is said to be edge covered critical if it is of class and every edge with vertices in V(G) not belonging to E(G) is critical. Some properties about edge covered critical graphs are considered.     

On cyclic edge-connectivity of transitive graphs

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is said to be cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity cλ(G) is the cardinality of a minimum cyclic edge-cut of G. In this paper, we first prove that for any cyclically separable graph G, , where ω(X) is the number of edges with one end in X and the other end in V(G)?X. A cyclically separable graph G with cλ(G)=ζ(G) is said to be cyclically optimal. The main results in this paper are: any connected k-regular vertex-transitive graph with k≥4 and girth at least 5 is cyclically optimal; any connected edge-transitive graph with minimum degree at least 4 and order at least 6 is cyclically optimal.