Recognizing claw-free perfect graphs

AbstractWe present a polynomial-time algorithm to recognize claw-free perfect graphs. The algorithm is based on a decomposition theorem elucidating the structure of these graphs

Every Elementary Graph is Chromatic Choosable

Elementary graphs are graphs whose edges can be colored using two colors in such a way that the edges in any induced $P_3$ get distinct colors. They constitute a subclass of the class of claw-free perfect graphs. In this paper, we show that for any elementary graph, its list chromatic number and chromatic number are equal

Eulerian and Hamiltonian properties of Gallai and anti-Gallai middle graphs

The Gallai middle graph ΓM(G) of a graph G = (V, E) is the graph whose vertex set is V ∪ E and two edges ei, ej ∈ E are adjacent in ΓM(G), if they are adjacent edges of G and do not lie on a same triangle in G, or if ei = uv ∈ E then ei is adjacent to u and v in ΓM(G). The anti-Gallai middle graph ∆M(G) of a graph G = (V, E) is the graph whose vertex set is V ∪ E and two edges ei, ej ∈ E are adjacent in ∆M(G) if they are adjacent in G and lie on a same triangle in G, or if ei = uv ∈ E then ei is adjacent to u and v in ∆M(G). In this paper, we investigate Eulerian and Hamiltonian properties of Gallai and anti-Gallai middle graphs.Publisher's Versio

Some properties of graphs determined by edge zeta functions

AbstractIn 1989, Hashimoto introduced an edge zeta function of a finite graph, which is a generalization of the Ihara zeta function. The edge zeta function is the reciprocal of a polynomial in twice as many indeterminants as edges in the graph and can be computed via a determinant expression. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal. Finally, we present a new example illustrating that the Ihara zeta function cannot necessarily do the same