We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., subdivided to a d-dimensional geometric cross-polytopal complex. give positive answer in dimension 3, with additional property that octahedralization introduces no new vertices on boundary polytope.

Let G be a graph with n vertices and suppose that for each vertex v in G, 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. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in character...

Let G be a graph with ν vertices, and let S1, S2, . . . , Sν be a list of colors on its vertices, each of size k. If there exists a unique proper coloring for G from this list of colors, then G is called uniquely k–list colorable graph. We characterize all uniquely 2–list colorable graphs, and discuss uniquely k–list colorable graphs by introducing some open problems. We also show the connectio...

Let G be a graph with n vertices and suppose that for each vertex v in G, 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. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in character...

Let G be a graph with vertices, and let S 1 ; S 2 ; : : : ; S be a list of colors on its vertices, each of size k. If there exists a unique proper coloring for G from this list of colors, then G is called uniquely k{list colorable graph. We prove that a graph is uniquely 2{list colorable if and only if one of its blocks is not a cycle, a complete graph, or a complete bipartite graph. For each k...

This paper studies the concepts of uniquely colorable graphs & Perfect graphs. The main results are 1) Every uniquely k-colorable graph is (k 1)-connected. 2) If G is a uniquely k-colorable graph, then  (G) ≥ k l. 3) A maximal planar graph G of order 3 or more has chromatic number 3 if and only if G is Eulerian. 4) Every interval graph is perfect. 5) A graph G is chordal if and only if G can b...

We consider the problem of testing 3-colorability in the bounded-degree model. A 3-colorability tester is an algorithm A that is given oracle access to the adjacency list representation of a graph G of maximum degree d with n vertices; A is required to, say, accept with probability at least 2/3 if G is 3-colorable, and to accept with probability at most 1/3 if G is -far from 3-colorable (meanin...