Color-bounded hypergraphs, I: General results

The concept of color-bounded hypergraph is introduced here. It is a hypergraph (set system) with vertex set X and edge set E={E₁,…,E_m}, where each edge E_i is associated with two integers s_i and t_i such that 1≤s_i≤t_i≤|E_i|. A vertex coloring φ:X→N is considered to be feasible if the number of colors occurring in E_i satisfies s_i≤|φ(E_i)|≤t_i, for all i≤m.Color-bounded hypergraphs generalize the concept of ‘mixed hypergraphs’ introduced by Voloshin [V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45-52], and a recent model studied by Drgas-Burchardt and ?azuka [E. Drgas-Burchardt, E. ?azuka, On chromatic polynomials of hypergraphs, Applied Mathematics Letters 20 (12) (2007) 1250-1254] where only lower bounds s_i were considered.We discuss the similarities and differences between our general model and the more particular earlier ones. An important issue is the chromatic spectrum-strongly related to the chromatic polynomial-which is the sequence whose kth element is the number of allowed colorings with precisely k colors (disregarding color permutations). Problems concerning algorithmic complexity are also considered.     

Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set E={E₁,…,E_m}, together with integers s_i and t_i (1≤s_i≤t_i≤|E_i|) for i=1,…,m. A vertex coloring φ is feasible if the number of colors occurring in edge E_i satisfies s_i≤|φ(E_i)|≤t_i, for every i≤m.In this paper we point out that hypertrees-hypergraphs admitting a representation over a (graph) tree where each hyperedge E_i induces a subtree of the underlying tree-play a central role concerning the set of possible numbers of colors that can occur in feasible colorings. We also consider interval hypergraphs and circular hypergraphs, where the underlying graph is a path or a cycle, respectively. Sufficient conditions are given for a ‘gap-free’ chromatic spectrum; i.e., when each number of colors is feasible between minimum and maximum. The algorithmic complexity of colorability is studied, too.Compared with the ‘mixed hypergraphs’-where ‘D-edge’ means (s_i,t_i)=(2,|E_i|), while ‘C-edge’ assumes (s_i,t_i)=(1,|E_i|−1)-the differences are rather significant.     

Upper bounds on vertex distinguishing chromatic index of some Halin graphs

A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by ???? _s (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with ??(G) ?? 4, respectively.     

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.     

Concise proofs for adjacent vertex-distinguishing total colorings

Let G=(V,E) be a graph and f:(V∪E)→[k] be a proper total k-coloring of G. We say that f is an adjacent vertex- distinguishing total coloring if for any two adjacent vertices, the set of colors appearing on the vertex and incident edges are different. We call the smallest k for which such a coloring of G exists the adjacent vertex-distinguishing total chromatic number, and denote it by χ_at(G). Here we provide short proofs for an upper bound on the adjacent vertex-distinguishing total chromatic number of graphs of maximum degree three, and the exact values of χ_at(G) when G is a complete graph or a cycle.     

The edge-face coloring of graphs embedded in a surface of characteristic zero

Let G be a graph embedded in a surface of characteristic zero with maximum degree Δ. The edge-face chromatic number χ_ef(G) of G is the least number of colors such that any two adjacent edges, adjacent faces, incident edge and face have different colors. In this paper, we prove that χ_ef(G)≤Δ+1 if Δ≥13, χ_ef(G)≤Δ+2 if Δ≥12, χ_ef(G)≤Δ+3 if Δ≥4, and χ_ef(G)≤7 if Δ≤3.     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.     

The strong chromatic index of a class of graphs

The strong chromatic index of a graph G is the minimum integer k such that the edge set of G can be partitioned into k induced matchings. Faudree et al. [R.J. Faudree, R.H. Schelp, A. Gyárfás, Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205-211] proposed an open problem: If G is bipartite and if for each edge xy∈E(G), d(x)+d(y)≤5, then sχ^′(G)≤6. Let H₀ be the graph obtained from a 5-cycle by adding a new vertex and joining it to two nonadjacent vertices of the 5-cycle. In this paper, we show that if G (not necessarily bipartite) is not isomorphic to H₀ and d(x)+d(y)≤5 for any edge xy of G then sχ^′(G)≤6. The proof of the result implies a linear time algorithm to produce a strong edge coloring using at most 6 colors for such graphs.     

Fractional weak discrepancy and split semiorders

The fractional weak discrepancywd_F(P) of a poset P=(V,?) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative k for which there exists a function f:V→R satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5] and [7] on the range of wd_F for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0≤s−1≤r<2s.     

Some results in square-free and strong square-free edge-colorings of graphs

The set of problems we consider here are generalizations of square-free sequences [A. Thue, Über unendliche Zeichenreichen, Norske Vid Selsk. Skr. I. Mat. Nat. Kl. Christiana 7 (1906) 1-22]. A finite sequence a₁a₂…a_n of symbols from a set S is called square-free if it does not contain a sequence of the form ww=x₁x₂…x_mx₁x₂…x_m,x_i∈S, as a subsequence of consecutive terms. Extending the above concept to graphs, a coloring of the edge set E in a graph G(V,E) is called square-free if the sequence of colors on any path in G is square-free. This was introduced by Alon et al. [N. Alon, J. Grytczuk, M. Ha?uszczak, O. Riordan, Nonrepetitive colorings of graphs, Random Struct. Algor. 21 (2002) 336-346] who proved bounds on the minimum number of colors needed for a square-free edge-coloring of G on the class of graphs with bounded maximum degree and trees. We discuss several variations of this problem and give a few new bounds.     

The deficiency of a regular graph

A proper edge coloring c:E(G)→Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1?k?4 and bounds for larger k.     

Balanced decomposition of a vertex-colored graph

A balanced vertex-coloring of a graph G is a function c from V(G) to {−1,0,1} such that ∑{c(v):v∈V(G)}=0. A subset U of V(G) is called a balanced set if U induces a connected subgraph and ∑{c(v):v∈U}=0. A decomposition V(G)=V₁∪?∪V_r is called a balanced decomposition if V_i is a balanced set for 1≤i≤r.In this paper, the balanced decomposition number f(G) of G is introduced; f(G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decomposition V(G)=V₁∪?∪V_r with |V_i|≤s for 1≤i≤r. Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.     

The skew chromatic index of a graph

We define a skew edge coloring of a graph to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. We show that this concept is closely related to that of skew Room squares and use this relation to prove that s(G) is at most o(G) + 4. We also find better upper bounds for s(G) when G is cyclic, cubic, or bipartite. In particular we use a construction involving Latin squares to show that if G is complete bipartite of order 2n, s(G) is bounded above by roughly $\text{3n}\text{2}$.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

Rainbows in the Hypercube

Let Q_n be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples differ in exactly one position. An edge coloring of a graph H is called rainbow if no two edges of H have the same color. Let f(G,H) be the largest number of colors such that there exists an edge coloring of G with f(G,H) colors such that no subgraph isomorphic to H is rainbow. In this paper we start the investigation of this anti-Ramsey problem by providing bounds on f(Q_n,Q_k) which are asymptotically tight for k = 2 and by giving some exact results.     

On the Ramsey multiplicity for stars

In 1929, Ramsey proved a theorem guaranteeing that if G₁,G₂,…,G_k are graphs, then there exists an integer r so that if the edges of K_r are colored in any fashion with k colors a monochromatic G_i in color i exists for some i. Harary and Prins suggested the problem of deciding the minimum number of monochromatic G_i in any such coloring. It is the purpose of this paper to establish this minimum number in the case when G_i are stars for each i.     

Acyclic edge coloring of subcubic graphs

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 it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

A generalization of Fan’s results: Distribution of cycle lengths in graphs

Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(C_k)|−|E(C_k−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if d_G(u)+d_G(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if d_G(u)+d_G(v)≥6k+1, then |E(C_k)|−|E(C_k−1)|=2.     

Perfect difference systems of sets and Jacobi sums

A perfect (v,{k_i∣1≤i≤s},ρ) difference system of sets (DSS) is a collection of s disjoint k_i-subsets D_i, 1≤i≤s, of any finite abelian group G of order v such that every non-identity element of G appears exactly ρ times in the multiset {a−b∣a∈D_i,b∈D_j,1≤i≠j≤s}. In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection {D_i∣1≤i≤s} defined in a finite field F_q of order q=ef+1 to be a perfect (q,{k_i∣1≤i≤s},ρ)-DSS, where each D_i is a union of cyclotomic cosets of index e (and the zero 0∈F_q). Also, we give numerical results for the cases e=2,3, and 4.