2 votes 2 votes Which of the following is/are correct? $S_1:$ A connected graph of at least $2$ vertices has an Euler circuit if and only if degree of every vertex is even. $S_2:$ A connected graph has an Euler path but not an Euler circuit if and only if there are at most two vertices with odd degree. $S_1$ only $S_2$ only Both $S_1$ and $S_2$ Neither $S_1$ nor $S_2$ Graph Theory goclasses2024-iiith-mock-5 goclasses graph-theory graph-connectivity euler-graph 1-mark + – GO Classes asked Apr 30, 2023 • retagged Apr 29 by Lakshman Bhaiya GO Classes 210 views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
2 votes 2 votes $S_2$ is actually not TRUE as it is “iff” and not “if”. A connected graph has an Euler path but not Euler circuit if and only if there are exactly two vertices of odd degree. So, the correct answer is $(A)$. GO Classes answered Apr 30, 2023 • edited Sep 11, 2023 by Deepak Poonia GO Classes comment Share Follow See 1 comment See all 1 1 comment reply okntk commented May 1, 2023 reply Follow Share So, could an Eulerian path exist even if $\#v_{odd-degree}>2$? I don’t think I have studied this anywhere though. 0 votes 0 votes Please log in or register to add a comment.
