We show that any connected regular graph with d + 1 distinct eigenvalues and odd-girth 2d + 1 is distance-regular, and in particular that it is a generalized odd graph. 2010 Mathematics Subject Classification: 05E30, 05C50; JEL Classification System: C0

We study a variant of intersection representations with unit balls, that is, unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far sets, the goal is to represent the vertices of the graph by unit balls so that the balls representing two adjacent vertices intersect if and only if the corresponding edge is near. ...

In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q ≥ k. Examining the construction method, we prove that starting from PG(2, q), q = p, p prime, there are no other constructions using this idea resulting in a (q + 1− t)-regular graph of girth six than the known ones, if t is not too large (t ≤ p and roug...

As an extension of the Four-Color Theorem it is conjectured that every planar graph of odd-girth at least 2k + 1 admits a homomorphism to PC2k = (Z 2 , {e1, e2, · · · , e2k, J}) where ei’s are standard basis and J is all 1 vector. Noting that PC2k itself is of odd-girth 2k + 1, in this work we show that if the conjecture is true, then PC2k is an optimal such a graph both with respect to number ...

Robertson has conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumpti...

A graph is superconnected, for short super-κ, if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any r-regular graph of diameter D and odd girth g that if D ≤ g − 2, then the graph is super-κ when g ≥ 5 and a complete graph otherwise.

We show that for every ε > 0 there exists an r0 = r0(ε) such that for all integers r ≥ r0 every graph of average degree at least r + ε and girth at least 1000 contains a subdivision of Kr+2. Combined with a result of Mader this implies that for every ε > 0 there exists an f(ε) such that for all r ≥ 2 every graph of average degree at least r + ε and girth at least f(ε) contains a subdivision of ...

We give here some new lower bounds on the order of a largest induced forest in planar graphs with girth 4 and 5. In particular we prove that a triangle-free planar graph of order n admits an induced forest of order at least 6n+7 11 , improving the lower bound of Salavatipour [M. R. Salavatipour, Large induced forests in trianglefree planar graphs, Graphs and Combinatorics, 22:113–126, 2006]. We...

It is well known that for any k and g, there is a graph with chromatic number at least k and girth at least g. In 1970’s, Erdős and Hajnal conjectured that for any numbers k and g, there exists a number f(k, g), such that every graph with chromatic number at least f(k, g) contains a subgraph with chromatic number at least k and girth at least g. In 1978, Rödl proved the case for g = 4 and arbit...