A characterization of canonically consistent total signed graphs

Deepa Sinha and Pravin Garg
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 3, Pages 70–77
Full paper (PDF, 164 Kb)


Authors and affiliations

Deepa Sinha
Department of Mathematics, South Asian University, Akbar Bhawan
Chanakyapuri, New Delhi–110021, India

Pravin Garg
Centre for Mathematical Sciences, Banasthali University
Banasthali–304022, Rajasthan, India


The canonical marking on a signed graph (or sigraph, in short) S is defined as: for each vertex v ∈ V (S), μσ(v) = Πej ∈ Ev, where Ev is the set of edges ej incident at v in S. If S is canonically marked, then a cycle Z in S is said to be canonically consistent (C-consistent) if it contains an even number of negative vertices and the given sigraph S is C-consistent if every cycle in it is C-consistent. The total sigraph T(S) of a sigraph S = (V, E, σ) has T(Se) as its underlying graph and for any edge uv of T(Se),
\sigma_T(uv) = \begin{cases} \sigma(uv) & \text{if \ $u,v \in V,$}\\ \sigma(u)\sigma(v) & \text{if \ $u,v \in E,$}\\ \sigma(u)\prod_{e_j \in E_{v}} \sigma(e_j) & \text{if \ $u \in E \ $and$ \ v \in V.$} \end{cases}
In this paper, we establish a characterization of canonically consistent total sigraphs.


  • Sigraph
  • Canonical marking
  • Consistent sigraph
  • Total sigraph

AMS Classification

  • 05C22
  • 05C75


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Cite this paper

Sinha, D., & Garg, P. (2013). A characterization of canonically consistent total signed graphs. Notes on Number Theory and Discrete Mathematics, 19(3), 70-77.

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