We show that there exist k-colorable matroids are not (b,c)-decomposable when b and c constants. A matroid is (b,c)-decomposable, if its ground set of elements can be partitioned into sets X1,X2,…,Xℓ with the following two properties. Each Xi has size at most ck. Moreover, for all Y such |Y∩Xi|≤1 it case b-colorable. (b,c)-decomposition a strict generalization partition decomposition and, thus,...

Given an undirected node-weighted graph and a positiveinteger k, the maximum k-colorable subgraph problem is to select ak-colorable induced subgraph of largest weight. The natural integerprogramming formulation for this problem exhibits a high degree ofsymmetry which arises by permuting the color classes. It is well knownthat such symmetry has negative effects on the perform...

In simple graphs, DP-coloring is a generalization of list coloring and thus many results generalize those coloring. Xu Wu proved that every planar graph without 5-cycles adjacent simultaneously to 3-cycles 4-cycles 4-choosable. Later, Sittitrai Nakprasit showed if has no pairwise 3-, 4-, 5-cycles, then it DP-4-colorable, which the result Wu. this paper, we extend on 5-, 6-cycles by showing 3-cy...

Question 1 Proposition 1. A graph X is 2-colorable if and only if it is bipartite. Proof. If a graph X is 2-colorable, we can color it using colors a and b. Grouping vertices of color a in A and those of color b in B, we obtain two sets inside which no two vertices are adjacent. Conversly, if X is Bipartite, we can group the vertices in sets A and B. Simply color all vertices of A with color a ...

For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: K2,2,r r ∈ {4, 5, 6, 7, 8}, K2,3,4, K1∗4,4, K1∗4,5, K1∗5,4. Also, they con...

A proper edge-coloring of a graph G with colors 1, . . . , t is called an interval t-coloring if the colors of edges incident to any vertex of G form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. For an interval colorable graph G, the least value of t for which G has an interval t-coloring is denoted by w(G). A graph G is ...

For a mixed hypergraphH= (X,C,D), where C andD are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’such that everyC ∈ Cmeets some class in more than one vertex, and everyD ∈ D has a nonempty intersection with at least two classes.A vertex-order x1, x2, . . . , xn onX (n=|X|) is uniquely colorable if the subhypergraph induced by {xj : 1 j i} has precisely on...