Triangle-free regular graphs

Triangle-free distance-regular graphs

Let Γ = (X, R) denote a distance-regular graph with distance function ∂ and diameter d ≥ 3. For 2 ≤ i ≤ d, by a parallelogram of length i, we mean a 4-tuple xyzu of vertices in X such that ∂(x, y) = ∂(z, u) = 1, ∂(x, u) = i, and ∂(x, z) = ∂(y, z) = ∂(y, u) = i − 1. Suppose the intersection number a1 = 0, a2 6= 0 in Γ. We prove the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial and con...

Deficiencies of connected regular triangle free graphs

Let G be a finite simple graph having a maximum matching M. The deficiency def(G) of G is the number of M-unsaturated vertices in G. In an earlier paper we determined an upper bound for def(G) when G is regular and connected. This upper bound is in general not sharp when G is triangle free. In this paper we study the case when G is triangle free and r-regular. We present an upper bound for defC...

Decomposing 4-Regular Graphs into Triangle-Free 2-Factors

There is a polynomial algorithm which finds a decomposition of any given 4-regular graph into two triangle-free 2-factors or shows that such a decomposition does not exist.

Measuring triangle-free graphs

Median Graphs and Triangle-Free Graphs

Let M(m,n) be the complexity of checking whether a graph G with m edges and n vertices is a median graph. We show that the complexity of checking whether G is triangle-free is at most M(m,m). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most O(m log n) + T (m log n, n), where T (m,n) is the complexity of finding all triangles of the graph. W...