The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some ...

The simplest graph drawing method is that of putting the vertices of a graph on a line (spine) and drawing the edges as half-circles on k half planes (pages). Such drawings are called k-page book drawings and the minimal number of edge crossings in such a drawing is called the k-page crossing number. In a one-page book drawing, all edges are placed on one side of the spine, and in a two-page bo...

Poljak, S. and D. Turzik, Max-cut in circulant graphs, Discrete Mathematics 108 (1992) 379-392. We study the max-cut problem in circulant graphs C,,,, where C,,, is a graph whose edge set consists of a cycle of length n and all the vertex pairs of distance r on the cycle. An efficient solution of the problem is obtained so that we show that there is always a maximum cut of a particular shape, c...

For given positive integers n; a1; : : : ; am, we consider the undirected circulant graph G=(V; E) with set of vertices V = {0; : : : ; n − 1} and set of edges E = {[i; j]: i − j ≡ ±ak (mod n) for some 16 k6m}. We prove that G is planar if m = 1 and non-planar if m¿ 3. For m = 2 we completely characterize planarity. It is shown that G is bipartite if and only if there is an l such that 2 divide...

For any given graph $G$ consider a $\widetilde{G}$ which is cone over $G.$ In this paper, we study two important invariants of such cone. Namely, complexity (the number spanning trees) and the Jacobian graph. We prove that coincides rooted forests in isomorphic to cokernel operator $I+L(G),$ where $L(G)$ Laplacian $I$ identity matrix. As consequence, one can calculate as $\det(I+L(G)).$

The bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number.A dominating setD is called an efficient dominating set ofG if |N−[v]∩D|=1 for every vertex v ∈ V (G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the ...

In 2002, Tonchev first constructed some linear binary codes defined by the adjacency matrices of undirected graphs. So graph is an important tool for searching optimum code. In this paper, we introduce a new method of searching (proposed) optimum formally self-dual linear binary codes from circulant graphs. AMS Subject Classification 2010: 94B05, 05C50, 05C25.

We construct a polynomial-time algorithm that given a graph X with 4p vertices (p is prime), finds (if any) a Cayley representation of X over the group C2 × C2 × Cp. This result, together with the known similar result for circulant graphs, shows that recognising and testing isomorphism of Cayley graphs over an abelian group of order 4p can be done in polynomial time.

The matching preclusion set of a graph is a set of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. The matching preclusion number is the minimum cardinality over all matching preclusion sets. We show in this paper that, for any ≥  , the matching preclusion numbers of both -dimensional restricted HL-graph and recursive circulant  ...

Let M = } ,..., , { 2 1 n v v v be an ordered set of vertices in a graph G. Then )) , ( ),..., , ( ), , ( ( 2 1 n v u d v u d v u d is called the M-coordinates of a vertex u of G. The set M is called a metric basis if the vertices of G have distinct M-coordinates. A minimum metric basis is a set M with minimum cardinality. The cardinality of a minimum metric basis of G is called minimum metric ...