Local condition for planar graphs of maximum degree 7 to be 8-totally colorable

The total chromatic number of a graph G, denoted by χ^″(G), is the minimum number of colors needed to color the vertices and edges of G such that no two adjacent or incident elements get the same color. It is known that if a planar graph G has maximum degree Δ≥9, then χ^″(G)=Δ+1. In this paper, we prove that if G is a planar graph with maximum degree 7, and for every vertex v, there is an integer k_v∈{3,4,5,6} so that v is not incident with any k_v-cycle, then χ^″(G)=8.     

Acyclic chromatic indices of planar graphs with large girth

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree Δ has a^′(G)=Δ if there exists a pair (k,m)∈{(3,11),(4,8),(5,7),(8,6)} such that G satisfies Δ≥k and g(G)≥m.     

Type-B generalized triangulations and determinantal ideals

For n≥3, let Ω_n be the set of line segments between the vertices of a convex n-gon. For j≥2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k≥1, let Δ_n,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing.The complex Δ_n,k has been the central object of many papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line segments in Ω_2n which can be transformed into each other by a 180^°-rotation of the 2n-gon. Let F_n be the set Ω_2n after identification, then the complex D_n,k of type-B generalized triangulations is the simplicial complex of subsets of F_n not containing any (k+1)-crossing in the above sense. For k=1, we have that D_n,1 is the simplicial complex of type-B triangulations of the 2n-gon as defined in [R. Simion, A type-B associahedron, Adv. Appl. Math. 30 (2003) 2-25] and decomposes into a join of an (n−1)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that D_n,k is a pure, k(n−k)−1+kn dimensional complex that decomposes into a kn−1-simplex and a k(n−k)−1 dimensional homology-sphere. For k=n−2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of D_n,k.On the algebraical side we give a term order on the monomials in the variables X_ij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1) times (k+1) minors of the generic n×n matrix contains the Stanley-Reisner ideal of D_n,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k<n.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

deBruijn-like sequences and the irregular chromatic number of paths and cycles

A deBruijn sequence of orderk, or a k-deBruijn sequence, over an alphabet A is a sequence of length |A|^k in which the last element is considered adjacent to the first and every possible k-tuple from A appears exactly once as a string of k-consecutive elements in the sequence. We will say that a cyclic sequence is deBruijn-like if for some k, each of the consecutive k-element substrings is distinct.A vertex coloring χ:V(G)→[k] of a graph G is said to be proper if no pair of adjacent vertices in G receive the same color. Let C(v;χ) denote the multiset of colors assigned by a coloring χ to the neighbors of vertex v. A proper coloring χ of G is irregular if χ(u)=χ(v) implies that C(u;χ)≠C(v;χ). The minimum number of colors needed to irregularly color G is called the irregular chromatic number of G. The notion of the irregular chromatic number pairs nicely with other parameters aimed at distinguishing the vertices of a graph. In this paper, we demonstrate a connection between the irregular chromatic number of cycles and the existence of certain deBruijn-like sequences. We then determine the exact irregular chromatic number of C_n and P_n for n≥3, thus verifying two conjectures given by Okamoto, Radcliffe and Zhang.     

Linear choosability of graphs

A proper vertex coloring of a non-oriented graph G is linear if the graph induced by the vertices of any two color classes is a forest of paths. A graph G is linearly L-list colorable if for a given list assignment L={L(v):v∈V(G)}, there exists a linear coloring c of G such that c(v)∈L(v) for all v∈V(G). If G is linearly L-list colorable for any list assignment with |L(v)|?k for all v∈V(G), then G is said to be linearly k-choosable. In this paper, we investigate the linear choosability for some families of graphs: graphs with small maximum degree, with given maximum average degree, outerplanar and planar graphs. Moreover, we prove that deciding whether a bipartite subcubic planar graph is linearly 3-colorable is an NP-complete problem.     

Sparse connectivity certificates via MA orderings in graphs

For an undirected multigraph G=(V,E), let α be a positive integer weight function on V. For a positive integer k, G is called (k,α)-connected if any two vertices u,v∈V remain connected after removal of any pair (Z,E^′) of a vertex subset Z⊆V-{u,v} and an edge subset E^′⊆E such that ∑_v∈Zα(v)+|E^′|<k. The (k,α)-connectivity is an extension of several common generalizations of edge-connectivity and vertex-connectivity. Given a (k,α)-connected graph G, we show that a (k,α)-connected spanning subgraph of G with O(k|V|) edges can be found in linear time by using MA orderings. We also show that properties on removal cycles and preservation of minimum cuts can be extended in the (k,α)-connectivity.     

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) 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). Let Δ=Δ(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K_n,n. Alon, McDiarmid and Reed observed that a^′(K_p−1,p−1)=p for every prime p. In this paper we prove that a^′(K_p,p)≤p+2=Δ+2 when p is prime. Basavaraju, Chandran and Kummini proved that a^′(K_n,n)≥n+2=Δ+2 when n is odd, which combined with our result implies that a^′(K_p,p)=p+2=Δ+2 when p is an odd prime. Moreover we show that if we remove any edge from K_p,p, the resulting graph is acyclically Δ+1=p+1-edge-colorable.     

The Sigma Chromatic Number of a Graph

For a nontrivial connected graph G, let ${c: V(G)\to {{\mathbb N}}}For a nontrivial connected graph G, let c: V(G)? \mathbb N{c: V(G)\to {{\mathbb N}}} be a vertex coloring of G, where adjacent vertices may be colored the same. For a vertex v of G, let N(v) denote the set of vertices adjacent to v. The color sum σ(v) of v is the sum of the colors of the vertices in N(v). If σ(u) ≠ σ(v) for every two adjacent vertices u and v of G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number σ(G). The sigma chromatic number of a graph G never exceeds its chromatic number χ(G) and for every pair a, b of positive integers with a ≤ b, there exists a connected graph G with σ(G) = a and χ(G) = b. There is a connected graph G of order n with σ(G) = k for every pair k, n of positive integers with k ≤n if and only if k ≠ n − 1. Several other results concerning sigma chromatic numbers are presented.     

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 .     

Incidence coloring of the squares of some graphs

The incidence chromatic number of G, denoted by χ_i(G), is the least number of colors such that G has an incidence coloring. In this paper, we determine the incidence chromatic number of the powers of paths, trees, which are min{n,2k+1}, and Δ(T²)+1, respectively. For the square of a Halin graph, we give an upper bound of its incidence chromatic number.     

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.     

Relaxed very asymmetric coloring games

This paper investigates a competitive version of the coloring game on a finite graph G. An asymmetric variant of the (r,d)-relaxed coloring game is called the (r,d)-relaxed (a,b)-coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph G, using colors from a set X, with |X|=r. On each turn Alice colors a vertices and Bob colors b vertices. A color α∈X is legal for an uncolored vertex u if by coloring u with color α, the subgraph induced by all the vertices colored with α has maximum degree at most d. Each player is required to color an uncolored vertex legally on each move. The game ends when there are no remaining uncolored vertices. Alice wins the game if all vertices of the graph are legally colored, Bob wins if at a certain stage there exists an uncolored vertex without a legal color. The d-relaxed (a,b)-game chromatic number, denoted by , of G is the least r for which Alice has a winning strategy in the (r,d)-relaxed (a,b)-coloring game.The (r,d)-relaxed (1,1)-coloring game has been well studied and there are many interesting results. For the (r,d)-relaxed (a,1)-coloring game, this paper proves that if a graph G has an orientation with maximum outdegree k and a≥k, then for all d≥k²+2k; If a≥k³, then (a,1)- for all d≥2k+1.     

Coloring graphs from random lists of size 2

Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring φ such that φ(v)∈L(v) for all v∈V(G). In particular, we show that if g is odd and σ(n)=ω(n^1/(2g−2)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n→∞. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n≥g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if σ(n)=o(n^1/(2g−2)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n→∞. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ(n), exhibits a sharp threshold at σ(n)=2n.     

Total chromatic number of graphs with small genus

Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with k colors. Denote χ_ve (G) the total chromatic number of G, and c(Σ) the Euler characteristic of a surfase Σ. In this paper, we prove that for any simple graph G which can be embedded in a surface Σ with Euler characteristic c(Σ), χ_ve (G) = Δ (G) + 1 if c(Σ) > 0 and Δ (G) ≥ 13, or, if c(Σ) = 0 and Δ (G) ≥ 14. This result generalizes results in [3], [4], [5] by Borodin.     

Tournament immersion and cutwidth

A (loopless) digraph H is strongly immersed in a digraph G if the vertices of H are mapped to distinct vertices of G, and the edges of H are mapped to directed paths joining the corresponding pairs of vertices of G, in such a way that the paths used are pairwise edge-disjoint, and do not pass through vertices of G that are images of vertices of H. A digraph has cutwidth at most k if its vertices can be ordered {v₁,…,vn} in such a way that for each j, there are at most k edges uv such that u∈{v₁,…,vj−1} and v∈{vj,…,vn}.We prove that for every set S of tournaments, the following are equivalent:

•

there is a digraph H such that H cannot be strongly immersed in any member of S,

•

there exists k such that every member of S has cutwidth at most k,

•

there exists k such that every vertex of every member of S belongs to at most k edge-disjoint directed cycles.

This is a key lemma towards two results that will be presented in later papers: first, that strong immersion is a well-quasi-order for tournaments, and second, that there is a polynomial time algorithm for the k edge-disjoint directed paths problem (for fixed k) in a tournament.     

Colorful flowers

For a set A let k[A] denote the family of all k-element subsets of A. A function f:k[A]→C is a local coloring if it maps disjoint sets of A into different elements of C. A family F⊆k[A] is called a flower if there exists E∈[A]^k−1 so that |F∩F^′|=E for all F,F^′∈F, F≠F^′. A flower is said to be colorful if f(F)≠f(F^′) for any two F,F^′∈F. In the paper we find the smallest cardinal γ such that there exists a local coloring of k[A] containing no colorful flower of size γ. As a consequence we answer a question raised by Pelant, Holický and Kalenda. We also discuss a few results and conjectures concerning a generalization of this problem.     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.