On path-cycle decompositions of triangle-free graphs

Discrete Applied Mathematics, 2014

A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex v ∈ V (G). An (L, d) *-coloring is a mapping π that assigns a color π (v) ∈ L(v) to each vertex v ∈ V (G) so that at most d neighbors of v receive the color π (v). A graph G is said to be (k, d) *-choosable if it admits an (L, d) *-coloring for every list assignment L with |L(v)| ≥ k for all v ∈ V (G). In 2001, Lih et al. (2001) [6] proved that planar graphs without 4-and l-cycles are (3, 1) *-choosable, where l ∈ {5, 6, 7}. Later, Dong and Xu (2009) [3] proved that planar graphs without 4-and l-cycles are (3, 1) *-choosable, where l ∈ {8, 9}. There exist planar graphs containing 4-cycles that are not (3, 1) *-choosable (Cowen et al., 1986 [1]). This partly explains the fact that in all above known sufficient conditions for the (3, 1) *-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3-and 4-cycles is (3, 1) *-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang (2007) [11] and show that every planar graph without 4-cycles is (3, 1) *-choosable.

Discrete Mathematics, 2012

In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph G is 3-choosable if it is satisfied one of the following conditions: (1) each vertex x is neither incident to cycles of lengths 4, 9, ix with ix ∈ {5, 7, 8}, nor incident to 6-cycles adjacent to a 3-cycle. (2) each vertex x is not incident to cycles of lengths 4, 7, 9, ix with ix ∈ {5, 6, 8}. This work implies five results already published [13, 3, 7, 12, 4].

For the class of triangle-free graphs Brooks' Theorem can be restated in terms of forbidden induced subgraphs, i.e. let G be a triangle-free and K 1,r+1-free graph. Then G is r-colourable unless G is isomorphic to an odd cycle or a complete graph with at most two vertices. In this note we present an improvement of Brooks' Theorem for triangle-free and rsunshade-free graphs. Here, an r-sunshade (with r ≥ 3) is a star K 1,r with one branch subdivided. A classical result in graph colouring theory is the theorem of Brooks [2], asserting that every graph G is (∆(G))-colourable unless G is isomorphic to an odd cycle or a complete graph. Bryant [3] simplified this proof with the following characterization of cycles and complete graphs. Thereby he highlights the exceptional role of the cycles and complete graphs in Brooks' Theorem. Here we give a new elementary proof of this characterization. Proposition 1 (Bryant [3]). Let G be a 2-connected graph. Then G is a cycle or a complete graph if and only if G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Proof. Let G be a 2-connected graph of order n. If G is a cycle or a complete graph, then obviously G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Hence, assume that G is neither a cycle nor a complete graph and that G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Note that then there exists at least one vertex v of G with 2 < d G (v) < n − 1. Since G

Discrete Applied Mathematics, 2010

An acyclic coloring of a graph G is a coloring of its vertices such that : (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V (G). If G is acyclically L-list colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V (G), then G is acyclically k-choosable. In this paper, we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less 7 (resp. 5, 3, 2) is acyclically 3-choosable.

2015

The triangle graph of a graph G, denoted by T (G), is the graph whose vertices represent the triangles (K3 subgraphs) of G, and two vertices of T (G) are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of Cn-free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs G for which T (G) is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.

Journal of Graph Theory, 2008

We prove that a 171-edge-connected graph has an edgedecomposition into paths of length 3 if and only its size is divisible by 3. It is a long-standing problem whether 2-edge-connectedness is sufficient for planar triangle-free graphs, and whether 3-edge-connectedness suffices for graphs in general.

Discrete Mathematics, 2008

Some structural properties of planar graphs without 4-cycles are investigated. By the structural properties, it is proved that every planar graph G without 4-cycles is edge-(∆(G) + 1)-choosable, which perfects the result given by Zhang and Wu: If G is a planar graph without 4-cycles, then G is edge-t-choosable, where t = 7 if ∆(G) = 5, and otherwise t = ∆(G) + 1.

Ars Combinatoria -Waterloo then Winnipeg-

Dirac showed that a 2-connected graph of order n with minimum degree δ has circumference at least min{2δ, n}. We prove that a 2connected, triangle-free graph G of order n with minimum degree δ either has circumference at least min{4δ-4, n}, or every longest cycle in G is dominating. This result is best possible in the sense that there exist bipartite graphs with minimum degree δ whose longest cycles have length 4δ -4, and are not dominating.

Discrete Applied Mathematics, 2016

The triangle graph of a graph G, denoted by T (G), is the graph whose vertices represent the triangles (K 3 subgraphs) of G, and two vertices of T (G) are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of C n-free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs G for which T (G) is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.

Discrete Mathematics, 2010

Let G = (V , E) be a graph. A proper vertex coloring of G is acyclic if G contains no bicolored cycle. Namely, every cycle of G must be colored with at least three colors. G is acyclically L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper acyclic coloring π of G such that π (v) ∈ L(v) for all v ∈ V. If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V , then G is acyclically k-choosable. In this paper, we prove that planar graphs with neither {4, 5}-cycles nor 8-cycles having a triangular chord are acyclically 4-choosable. This implies that planar graphs either without {4, 5, 7}-cycles or without {4, 5, 8}-cycles are acyclically 4-choosable.