On a generalization of Ramsey theory

Pacific Journal of Mathematics, 1974

Let χ(G) denote the chromatic number of a graph G. For positive integers n ίf n 2 , , n k (k ^ 1) the chromatic Ramsey number χ(n ίf n 2 , , n k) is defined as the least positive integer p such that for any factorization K p-U*=i G if χ{G t) Ξ> w* for at least one i,l ^i ^k. It is shown that x(n u n 2 , , n k) = 1 + ΓR=i (n t-1). The vertex-arboricity a(G) of a graph G is the fewest number of subsets into which the vertex set of G can be partitioned so that each subset induces an acyclic graph. For positive integers n lt n 2 , , w* (ft ^ 1) the vertexarboricity Ramsey number a(n lt n 2 ,-,n k) is defined as the least positive integer p such that for any factorization K p-U*=i Gi> d(G t) ^ Ύii for at least one i, 1 rg % <Ξ k. It is shown that a(n lt n 2 , , n k) = 1 + 2k ΓR=i (n t-1).

Discrete Mathematics, 1989

Tke s&chromatic number X,(G) of a graph G = (V, E) is the smallest order k of a partition {v,, v,,-* * 9 V,} of the vertices V(G) such that the subgraph (V;:) induced by each subset V;: consists of a disjoint union of complete subgraphs. By definition, Xs(G) <X(G), the chromatic number of G. This paper develops properties of this lower bound for the chromatic number. 1. Introduction An n-coloring of a graph G = (V, E) is a function f from V onto N = 11 2 l l 9 n} such that whenever vertices u and v are adjacent, thenf(u) #f(v). Ai e'q;ivalent definition, which provides much of the motivation for this paper, is that an n-coloring of G is a partition {VI, V2,. .. , Vn} of the vertices V(G) into color classes, such that for every i = 1,2,. .. , n, the subgraph (6) induced by Vj is totally disconnected, or equivalently, is the disjoint union of K,'s (complete graphs with one vertex). A partition {V', V2,. .. , Vn} of V(G) is called complete ifforeveryi,j, lsi<lm < n, there exists a vertex u in & and a vertex u in I$ such that u and v are adjacent. The chromatic number X(G) and the achromatic number Y(G) are the smallest and largest integers n, respectively, for which G has a complete n-coloring. Notice that in the definition of the chromatic number the completeness of the partition is not required but follows easily from the definition, whereas the completeness of the partition used in defining the achromatic number is essential. The chromatic number, of course, is a very well studied parameter, whose his&y dates back to the famous Four Color Problem and the early work of Kempe in 1879 [25] and Heawood in 1890 [22]. The achromatic number was first studied as a parameter by Harary, Hedetniemi and Prins in 1967 [20], and later named and studied by Harary and Hedetniemi in 1970 [ 191. This paper was motivated in part by an interest in developing interesting lower d Research supported in part by Office of Naval Research Contract NOOO14-86-K-0693.

Graphs and Combinatorics, 2019

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we obtain structural characterization of connected {P 5 , K 4 , Kite, Bull}-free graphs which contains an induced C 5 and connected {P 6 , C 5 , K 1,3 }-free graphs that contains an induced C 6 . Also, we prove that {P 5 , K 4 , Kite, Bull}-free graphs that contains an induced C 5 and {P 6 , C 5 , P 5 , K 1,3 }free graphs which contains an induced C 6 are k-indicated colorable for all k ≥ χ(G). In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in .

Graphs and Combinatorics

In this paper we study three-color Ramsey numbers. Let K i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G 1 , G 2 and G 3 , if r(G 1 , G 2) ≥ s(G 3), then r(G 1 , G 2 , G 3) ≥ (r(G 1 , G 2) − 1)(χ(G 3) − 1) + s(G 3), where s(G 3) is the chromatic surplus of G 3 ; (ii)(k + m − 2)(n − 1) + 1 ≤ r(K 1,k , K 1,m , K n) ≤ (k + m − 1)(n − 1) + 1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m ≥ k ≥ 2, there is a constant c such that r(K k,m , K k,m , K n) ≤ c(n/ log n) k , and r(C 2m , C 2m , K n) ≤ c(n/ log n) m/(m−1) for sufficiently large n.

The Additive Coloring Problem is a variation of the Coloring Problem where labels of {1,. .. , k} are assigned to the vertices of a graph G so that the sum of labels over the neighborhood of each vertex is a proper coloring of G. The least value k for which G admits such labeling is called additive chromatic number of G. This problem was first presented by Czerwiński, Grytczuk anḋ Zelazny who also proposed a conjecture that for every graph G, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete graphs, trees, non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. In this work, we show that the conjecture holds for split graphs. We also present exact formulas for computing the additive chromatic number for some subfamilies of split graphs (complete split, headless spiders and complete sun), regular bipartite, complete multipartite, fan, windmill, circuit, wheel, cycle sun and wheel sun.

Journal of Combinatorial Theory, Series B, 1973

2008

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)| ≥ mS , where mS is the size 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. If χ (c) = χ (G), then c is called a minimum local coloring of G. The local coloring of graphs introduced by Chartrand et. al. in 2003. In this paper, following the study of this concept, first an upper bound for χ (G) where G is not complete graphs K4 and K5, is provided in terms of maximum degree Δ(G). Then the exact value of χ (G) for some special graphs G such as the cartesian product of cycles, paths and complete graphs is determined.

Discrete Mathematics

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χ pcf (G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χ pcf (G) in terms of other graph parameters. In particular, we show that χ pcf (G) ≤ 5∆(G) 2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4 ≤ k ≤ 6. The paper concludes with few open problems.

Proceedings of the American Mathematical Society, 1974

A sequence is called non-repetitive if no of its subsequences forms a repetition (a sequence r 1 , r 2 , . . . , r 2n such that r i = r n+i for all 1 ≤ i ≤ n). Let G be a graph whose vertices are coloured. A colouring ϕ of the graph G is non-repetitive if the sequence of colours on every path in G is non-repetitive. The Thue chromatic number, denoted by π(G), is the minimum number of colours of a non-repetitive colouring of G.