#### Title

### A study on oriented relative clique number

#### Article Type

Research Article

#### Publication Title

Discrete Mathematics

#### Abstract

An oriented graph is a directed graph with no directed cycle of length one or two. The relative clique number of an oriented graph is the cardinality of a largest subset X of vertices such that each pair of vertices is either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 32. We also prove an upper bound of 14 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number for the families of outerplanar graphs with girth at least g and planar graphs with girth at least g+2 for all g≥3. Moreover, we study the relation of oriented relative clique number with oriented chromatic number, oriented absolute clique number and maximum degree of a graph. We also show that oriented relative clique number of a connected subcubic graph is at most seven which weakly supports a conjecture by Sopena (JGT 1997).

#### First Page

2049

#### Last Page

2057

#### DOI

10.1016/j.disc.2018.04.001

#### Publication Date

7-1-2018

#### Recommended Citation

Das, Sandip; Prabhu, Swathy; and Sen, Sagnik, "A study on oriented relative clique number" (2018). *Journal Articles*. 1326.

https://digitalcommons.isical.ac.in/journal-articles/1326

## Comments

All Open Access, Bronze