A vertex subset $S$ of a graph $G$ is general position set if no lies on geodesic between two other vertices $S$. The cardinality largest the number (gp-number) ${\rm gp}(G)$ $G$. gp-number determined for some families Kneser graphs, in particular $K(n,2)$ and $K(n,3)$. sharp lower bound proved Cartesian products graphs. also joins coronas over line graphs complete

The Kneser graph K(m, n) has the n-subsets of {1, 2 . . . . . m} as its vertices, two such vertices being adjacent whenever they are disjoint. The kth multichromatic number of the graph G is the least integer t such that the vertices of G can be assigned k-subsets of {1, 2,... ,t}, so that adjacent vertices of G receive disjoint sets. The values of xk(K(m, n)) are computed for n = 2, 3 and boun...

In an earlier paper, the present authors (2013) [1] introduced the alternating chromatic number for hypergraphs and used Tucker’s Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the alternating chromatic number is a lower bound for the chromatic number. In this paper, we determine the chromatic number of some families of graphs by specifying their alternating...

A labeling f : V (G) → {1, 2, . . . , d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by χD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let χ(G) be the chromatic nu...

The Kneser graph K(n, r) has as vertices all r-subsets of an n-set with two vertices adjacent if the corresponding subsets are disjoint. It is conjectured that, except for K(5, 2), these graphs are Hamiltonian for all n ≥ 2r +1. In this note we describe an inductive construction which relates Hamiltonicity of K(2r + 2s, r) to Hamiltonicity of K(2r′+s, r′). This shows (among other things) that H...

In 1976 Stahl [13] de ned the m-tuple coloring of a graph G and formulated a conjecture on the multichromatic number of Kneser graphs. For m = 1 this conjecture is Kneser's conjecture which was solved by Lovász [10]. Here we show that Lovász's topological lower bound in this way cannot prove Stahl's conjecture. We obtain that the strongest index bound only gives the trivial m · ω(G) lower bound...

Chen [4] confirmed the Johnson-Holroyd-Stahl conjecture that the circular chromatic number of a Kneser graph is equal to its chromatic number. A shorter proof of this result was given by Chang, Liu, and Zhu [3]. Both proofs were based on Fan’s lemma [5] in algebraic topology. In this article we give a further simplified proof of this result. Moreover, by specializing a constructive proof of Fan...

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős–Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corres...

Weakening the notion of a strong (induced) matching of graphs, in this paper, we introduce the notion of a semistrong matching. A matching M of a graph G is called semistrong if each edge of M has a vertex, which is of degree one in the induced subgraph G1⁄2M . We strengthen earlier results by showing that for the subset graphs and for the Kneser graphs the sizes of the maxima of the strong and...

The base size of a permutation group, and the metric dimension of a graph, are both well-studied parameters which are closely related. We survey results on the relationship between the two, and with other, related parameters of groups, graphs, coherent configurations and association schemes. We also present some new results, including on the base sizes of wreath products in the product action, ...